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Spectral method

The spectral method is a class of numerical techniques used in applied mathematics and computational science to approximate solutions to partial differential equations (PDEs) by expanding the solution as a truncated series of global basis functions, such as Fourier series for periodic domains or orthogonal polynomials like Chebyshev or Legendre for non-periodic domains.^[1]^[2] These methods enforce the governing equations either through projection onto the basis (e.g., Galerkin approach) or by collocation at specific points, transforming the PDE into a system of ordinary differential equations (ODEs) that can be solved efficiently.^[2] Unlike local methods such as finite differences, spectral methods leverage the entire domain for approximation, achieving exponential convergence rates for smooth solutions, where the error decreases rapidly as the number of basis functions increases.^[3]^[1] Spectral methods are particularly advantageous for problems requiring high accuracy and resolution, such as those in fluid dynamics, where they demand significantly fewer degrees of freedom—approximately seven times fewer per dimension—compared to second-order finite difference schemes while delivering superior precision.^[3] The choice of basis functions is tailored to the problem's geometry and boundary conditions: Fourier bases excel in periodic settings due to their trigonometric nature, enabling straightforward handling of wave propagation and periodic flows, whereas polynomial bases like Chebyshev are preferred for bounded, non-periodic domains to manage boundary layers and singularities effectively.^[2] Error analysis in these methods typically involves estimates in Sobolev spaces, showing that for analytic solutions, the approximation error scales as O(N^{-m}) or better, where N is the polynomial degree and m reflects the solution's smoothness, often leading to machine-precision accuracy with modest N.^[2] Applications of spectral methods span diverse fields, including the simulation of incompressible Navier-Stokes equations for turbulent flows, the heat equation, the Korteweg-de Vries (KdV) equation for solitons, and Schrödinger equations in quantum mechanics.^[1]^[2] In engineering contexts, they are employed for aeroacoustics, electromagnetics, combustion modeling, and seismic wave propagation, benefiting from their efficiency in multi-dimensional problems and ability to resolve fine-scale features in smooth regimes.^[2] Implementations often rely on spectral-collocation or spectral-Galerkin formulations, with software tools like PseudoPack facilitating practical use in both one- and multi-dimensional settings.^[2] Despite their strengths, spectral methods are most effective for problems with sufficient smoothness, as discontinuities or sharp gradients can degrade convergence, necessitating hybrid approaches with local methods in such cases.^[3]

Fundamentals

Definition and core principles

Spectral methods constitute a class of numerical techniques employed in applied mathematics and scientific computing to solve partial differential equations (PDEs) by approximating the solution through an expansion in a basis of smooth, global functions that span the entire computational domain. Unlike local methods such as finite differences or finite elements, which rely on piecewise approximations over small subdomains, spectral methods represent the solution globally, leveraging the inherent smoothness of the basis functions to achieve high accuracy.^[4]^[5] The core principles of spectral methods center on the projection of the PDE onto a finite-dimensional subspace generated by these global basis functions, typically orthogonal polynomials like Chebyshev or Legendre, or trigonometric functions for periodic problems. This approximation takes the form u(x) \approx \sum_{k=0}^N \hat{u}_k \phi_k(x), where \{\phi_k(x)\} are the basis functions and \hat{u}_k are the expansion coefficients determined by techniques such as Galerkin projection or collocation. The method's efficiency stems from the use of fast transforms, such as the Fast Fourier Transform (FFT) for trigonometric bases, which enable rapid computation of derivatives and projections with near-linear scaling in the number of modes N.^[4]^[5]^[6] A key advantage of spectral methods lies in their convergence properties: for analytic or sufficiently smooth solutions, the approximation error decays exponentially with increasing N, often reaching machine precision with modest N values, such as around 20 modes for C^\infty functions on simple domains. This spectral accuracy arises because the global basis captures the solution's features across the domain without introducing artificial discontinuities, making the methods particularly effective for problems with smooth, periodic, or entire-domain behavior.^[6]^[3]

Historical development

The foundations of spectral methods lie in Joseph Fourier's 1822 publication of Théorie analytique de la chaleur, where he introduced Fourier series to decompose periodic functions into trigonometric components, enabling global approximations central to later spectral techniques. This approach provided the basis for expanding solutions of partial differential equations in eigenfunction series compatible with the problem's geometry. Complementing this, Boris Galerkin developed in 1915 a weighted residual method for solving variational problems in elasticity, such as beam and plate equilibria, which evolved into the spectral Galerkin framework for projecting differential equations onto finite-dimensional subspaces. Following World War II, spectral methods gained traction in meteorology and aerodynamics for simulating complex flows. Cornelius Lanczos advanced the field in his 1956 book Applied Analysis, exploring Fourier series and orthogonal expansions for practical computational problems in physics and engineering. A major breakthrough came in 1971 when Steven Orszag applied pseudospectral methods to model turbulence, using Fourier transforms to handle nonlinear interactions efficiently while mitigating aliasing errors through techniques like the 2/3-rule filtering.^[7] Key milestones in the 1960s and 1970s further propelled spectral methods toward practicality. The fast Fourier transform (FFT) algorithm, introduced by James Cooley and John Tukey in 1965, dramatically reduced the computational complexity of Fourier-based expansions from O(N²) to O(N log N), making high-resolution simulations feasible. In the 1970s, David Gottlieb and Steven Orszag extended these ideas to Chebyshev polynomial bases for non-periodic domains, as detailed in their 1977 monograph Numerical Analysis of Spectral Methods: Theory and Applications, which analyzed convergence and stability for fluid dynamics problems.^[1] By the 1980s, spectral methods saw widespread adoption in computational fluid dynamics (CFD), solidified by the comprehensive text Spectral Methods in Fluid Dynamics by Claudio Canuto, M. Yousuff Hussaini, Alfo Quarteroni, and Thomas A. Zang in 1988, which integrated theory with implementation strategies.^[8] From the 1990s onward, spectral methods evolved to leverage parallel computing architectures, enabling large-scale simulations on distributed systems through techniques like spectral elements and domain decomposition.^[9] High-performance implementations addressed challenges in multidomain and non-periodic settings, with ongoing refinements focusing on efficiency for turbulent flows and complex geometries in aerodynamics and beyond.

Mathematical foundations

Choice of basis functions

In spectral methods, the choice of basis functions is crucial for achieving high-order accuracy in approximating solutions to differential equations. For problems on periodic domains, trigonometric functions, such as sines and cosines, serve as the primary basis due to their natural alignment with periodic boundary conditions and their ability to represent smooth periodic functions efficiently.^[10] These functions form a complete orthogonal set in the space of square-integrable periodic functions, enabling exponential convergence rates for sufficiently smooth solutions.^[2] For non-periodic domains, polynomial bases are preferred, with Chebyshev polynomials of the first kind, defined as T_n(x) = \cos(n \arccos x) for x \in [-1, 1], being a widely adopted choice owing to their minimax properties and clustering of roots near the boundaries, which aids in boundary layer resolution.^[10] Legendre polynomials, denoted P_n(x), provide an alternative for non-periodic problems, particularly when uniform weighting in the inner product is desirable, as they are orthogonal with respect to the standard L^2 inner product on [-1, 1].^[2] Both families ensure spectral accuracy, where the error decreases faster than any power of the polynomial degree for analytic functions.^[11] The orthogonality of these bases is fundamental to their efficacy; for Chebyshev polynomials, they satisfy \int_{-1}^{1} T_m(x) T_n(x) (1 - x^2)^{-1/2} \, dx = 0 for m \neq n, with the weight function (1 - x^2)^{-1/2} reflecting the Chebyshev measure.^[10] Legendre polynomials are orthogonal under \int_{-1}^{1} P_m(x) P_n(x) \, dx = 0 for m \neq n, normalized such that the integral equals $2/(2n+1) on the diagonal.^[2] Completeness in the L^2 space guarantees that any function in the space can be approximated arbitrarily well by finite expansions of these bases.^[11] Selection criteria emphasize matching the basis to the problem's domain characteristics: trigonometric bases excel for periodic settings to avoid Gibbs phenomena at boundaries, while polynomials suit bounded non-periodic domains and yield spectral accuracy for smooth functions, often outperforming finite differences by orders of magnitude in convergence rate.^[10] Boundary conditions are handled through techniques like the tau method, which enforces constraints by modifying the highest-mode coefficients, or penalty methods that add stabilizing terms without altering the basis orthogonality.^[2] Domain transformations are essential for adapting general geometries; non-interval domains are mapped to [-1, 1] via affine or more complex mappings to leverage polynomial bases, preserving smoothness to maintain high accuracy.^[10] In discrete implementations, aliasing arises from the nonlinear interaction of modes in trigonometric or polynomial expansions, necessitating de-aliasing strategies like the 3/2 rule to filter high-wavenumber artifacts and ensure stability.^[2]

Projection and discretization techniques

Spectral methods discretize partial differential equations (PDEs) by expanding the solution in a basis of global functions, such as Fourier or polynomials, and applying projection or interpolation techniques to convert the continuous problem into a discrete algebraic system. These techniques ensure high accuracy by exploiting the smoothness of the solution and the rapid convergence of the basis expansions. The choice of projection method influences the ease of implementation, handling of boundary conditions, and computational efficiency, with each approach leading to matrix equations whose properties determine the method's stability and conditioning.^[1] The Galerkin method is a weighted residual approach that projects the residual of the PDE onto the basis functions to enforce orthogonality in an inner product space. For a differential equation Lu = f with solution approximated as u_N = \sum_{k=0}^N a_k \phi_k, the residual R(u_N) = Lu_N - f satisfies \int R(u_N) \phi_j \, dx = 0 for j = 0, \dots, N, where \phi_j are the basis functions. This leads to a stiffness matrix A_{jk} = \int \phi_j L \phi_k \, dx, coupling the coefficients a_k through the differential operator L. The method is particularly effective for problems with periodic boundaries, as the integrals can be computed analytically for orthogonal bases, yielding sparse or diagonal mass matrices in Fourier cases. However, for non-periodic problems, numerical quadrature is required, increasing computational cost.^[1] In contrast, the collocation method, often termed pseudospectral, enforces the PDE exactly at a set of discrete points rather than through integrals, simplifying implementation for nonlinear terms. The approximation u_N(x_j) satisfies L u_N(x_j) = f(x_j) at collocation points x_j, typically chosen as Gauss-Lobatto quadrature nodes for polynomial bases to ensure accurate interpolation and differentiation. Derivatives are approximated using differentiation matrices D_{jk} \approx \frac{d \phi_k}{dx}(x_j), allowing the operator L to be represented as a matrix product applied to the vector of nodal values. For Chebyshev bases, these points cluster near boundaries, improving resolution of boundary layers. The method's efficiency stems from fast transform techniques for evaluating nonlinearities, though it requires careful point selection to avoid Runge phenomena.^[1] The tau method addresses boundary conditions in polynomial approximations by perturbing the PDE with tau parameters, avoiding full projection while maintaining spectral accuracy. In this approach, the solution is expanded as u_N = \sum_{k=0}^N a_k \phi_k, and the residual is projected onto lower-order basis functions, with the highest modes adjusted to satisfy boundary conditions exactly, introducing tau corrections like L u_N - f = \sum_{m=1}^M \tau_m \phi_{N-m+1}. This method is useful for non-periodic problems where exact boundary enforcement is crucial, as it decouples the interior approximation from boundary satisfaction without additional variables. It shares similarities with Galerkin but reduces the system size for boundary-constrained problems.^[1]^[12] Regardless of the projection technique, time-independent spectral discretizations generally reduce to matrix equations of the form A \mathbf{u} = \mathbf{f}, where \mathbf{u} collects the expansion coefficients or nodal values, and A incorporates the operator L in the chosen basis. The spectral radius of A, determined by the eigenvalues of the basis and operator, governs exponential convergence rates for smooth solutions, often achieving machine precision with modest N. However, conditioning of A deteriorates as O(N^4) for second-order operators in polynomial bases, necessitating preconditioning or iterative solvers for large N; stability analysis reveals that the methods are well-conditioned for elliptic problems but require time-step restrictions in hyperbolic or parabolic cases to control error growth.^[1]^[13]

Specific methods

Fourier-based spectral methods

Fourier-based spectral methods utilize trigonometric basis functions, specifically complex exponentials or sines and cosines, to approximate solutions of partial differential equations on periodic domains. For a function u(x) defined on [0, 2\pi] with periodic boundary conditions, the approximation takes the form

u_N(x) = \sum_{k=-N/2}^{N/2-1} \hat{u}_k e^{i k x},

where the Fourier coefficients \hat{u}_k are computed via the discrete Fourier transform of grid-point values at x_j = 2\pi j / N, j = 0, \dots, N-1. This expansion leverages the orthogonality of the basis functions to project the governing equations onto the spectral space, enabling high accuracy for smooth periodic functions with exponential convergence rates.^[14] Differentiation in Fourier-based methods is performed efficiently in spectral space by multiplying the coefficients by the wave number: the derivative u'(x) corresponds to coefficients i k \hat{u}_k, followed by an inverse transform to return to physical space. This approach avoids the slower convergence of finite difference schemes and preserves the structure of linear operators like the Laplacian, which becomes multiplication by -k^2. Energy conservation is ensured through Parseval's theorem, which states that for the periodic interval [-\pi, \pi],

\int_{-\pi}^{\pi} |u(x)|^2 \, dx = 2\pi \sum_{k} |\hat{u}_k|^2,

equating the L^2 norm in physical space to the \ell^2 norm of the coefficients, a property vital for simulating conservative systems like incompressible flows.^[1] Nonlinear terms, such as products in advection equations, introduce aliasing errors when evaluated in spectral space due to the convolution theorem, where high-wavenumber interactions fold back into lower modes. To mitigate this, the 3/2-rule dealiasing technique extends the grid to $3N/2 points for computing the product, padding with zeros in spectral space, and then truncates to retain only the original N modes, eliminating quadratic aliasing while increasing computational cost by a factor of about 1.5 in one dimension. This method, originally developed for turbulence simulations, maintains stability and accuracy for nonlinear problems without excessive mode truncation.^[10] The pseudospectral implementation combines the efficiency of grid-point evaluations for nonlinearities with spectral differentiation, relying on the fast Fourier transform (FFT) for forward and inverse transforms between physical and spectral spaces. The FFT algorithm achieves O(N \log N) complexity per transform, dramatically reducing the cost compared to direct summation O(N^2), and enables practical simulations at high resolutions on periodic domains. This approach, pioneered in the early 1970s, forms the backbone of many computational fluid dynamics codes for periodic problems.^[5]

Polynomial-based spectral methods

Polynomial-based spectral methods employ orthogonal polynomial bases, such as Chebyshev or Legendre polynomials, to approximate solutions of differential equations on non-periodic domains, particularly those involving boundaries. These methods are well-suited for boundary value problems where the solution is smooth but not periodic, offering high-order accuracy through global expansions. Unlike Fourier methods, which rely on trigonometric functions for periodic settings, polynomial approaches use bases orthogonal over finite intervals like [-1, 1], enabling efficient handling of irregular geometries via domain mapping.^[2] The Chebyshev spectral method utilizes Chebyshev polynomials of the first kind, T_n(x), defined by the recurrence relation T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) with T_0(x) = 1 and T_1(x) = x, orthogonal on [-1, 1] with weight (1 - x^2)^{-1/2}. In the collocation framework, interpolation nodes are chosen as the Chebyshev-Gauss-Lobatto points x_j = \cos(\pi j / N), for j = 0, 1, \dots, N, which cluster near the endpoints to resolve boundary layers effectively. Quadrature weights for integration at these nodes are \omega_j = \pi / N for $1 \leq j \leq N-1, and \omega_0 = \omega_N = \pi / (2N), facilitating accurate computation of inner products. Derivatives are computed via the differentiation matrix D, with entries D_{jj} = -x_j / (2(1 - x_j^2)) for interior points, and boundary-adjusted values like D_{00} = -(2N^2 + 1)/6, derived from Lagrange interpolation properties; higher derivatives follow by repeated matrix multiplication.^[15]^[10] The Legendre spectral method, in contrast, uses Legendre polynomials P_n(x), satisfying the recurrence (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x) with P_0(x) = 1 and P_1(x) = x, which are orthogonal on [-1, 1] with respect to the uniform weight function w(x) = 1, such that \int_{-1}^1 P_n(x) P_m(x) \, dx = \frac{2}{2n+1} \delta_{nm}. Collocation or Galerkin projections employ Gauss-Legendre-Lobatto quadrature points, including the endpoints x_0 = -1 and x_N = 1, with interior nodes as roots of the derivative P_N'(x); corresponding weights are \omega_j = \frac{2}{N(N+1) [P_N(x_j)]^2}. Expansion coefficients for a function u(x) in the basis are given by \hat{u}_n = \frac{2n+1}{2} \int_{-1}^1 u(x) P_n(x) \, dx, computable via the orthogonality integral divided by the norm \int_{-1}^1 P_n^2(x) \, dx = \frac{2}{2n+1}.^[1]^[6] Boundary conditions are imposed in polynomial spectral methods through either collocation, where values at nodes satisfy Dirichlet or Neumann constraints directly, or Galerkin frameworks, which weakly enforce conditions via modified basis functions. For Dirichlet conditions u(\pm 1) = g_{\pm}, basis functions like \phi_k(x) = T_k(x) - T_{k+2}(x) (Chebyshev) or \phi_k(x) = P_k(x) - P_{k+2}(x) (Legendre) are used to ensure \phi_k(\pm 1) = 0, reducing the expansion degree while preserving orthogonality; Neumann conditions u'(\pm 1) = h_{\pm} follow similarly by adjusting derivative representations, such as u'(x) = \sum_{k=0}^{N-1} (2k+1) \tilde{u}_k P_{k+1}(x) for Legendre. General mixed conditions a^{\pm} u(\pm 1) + b^{\pm} u'(\pm 1) = c^{\pm} are handled by incorporating constraint matrices into the system coefficients.^[2]^[16] Efficient integration in these methods often relies on Clenshaw-Curtis quadrature, which expands the integrand in Chebyshev series and evaluates at the same nodes x_j = \cos(\pi j / N), yielding weights via a fast cosine transform with accuracy comparable to Gauss quadrature for smooth functions on [-1, 1]. For smooth non-periodic functions, polynomial spectral methods achieve spectral accuracy, with approximation errors decaying exponentially as O(e^{-c N}) or O(e^{-c \sqrt{N}}) in the polynomial degree N, depending on analyticity in a Bernstein ellipse or similar region, far surpassing algebraic convergence of local methods.^[10]^[17]^[18]

Implementation strategies

Solving linear problems

Spectral methods for solving linear differential equations typically involve discretizing the spatial domain using a basis of global functions, such as Fourier or Chebyshev polynomials, to form a system of linear algebraic equations. The core approach relies on constructing differentiation matrices that approximate differential operators at collocation points. For instance, the first derivative operator is represented by a matrix D, where the approximate derivative of a vector of function values \mathbf{u} is D \mathbf{u}. Higher-order derivatives, like the second derivative D^2, are obtained by matrix powers or products, enabling the discretization of equations such as the Helmholtz problem (D^2 + \lambda I) \hat{\mathbf{u}} = \mathbf{f}, where I is the identity matrix and \hat{\mathbf{u}} denotes coefficients or collocated values.^[5]^[19] These linear systems are solved using direct methods for moderate dimensions, such as LU decomposition, which is efficient for small N (number of modes or points) up to around 1000, achieving spectral accuracy with errors decaying exponentially in N. For larger N, iterative solvers like GMRES or conjugate gradients are preferred, often preconditioned with incomplete LU factorizations or multigrid techniques to handle the ill-conditioning of differentiation matrices, whose condition numbers grow as O(N^2) for second derivatives. Multiplication by variable coefficients is incorporated via diagonal matrices or tensor products in multi dimensions, maintaining the linear structure. Recent advances include automated computation of adjoints for gradient-based optimization in spectral PDE solvers, improving efficiency in inverse problems and machine learning applications as of 2025.^[5]^[19]^[20] Eigenvalue problems, particularly Sturm-Liouville forms like -u'' + q(x)u = \lambda w(x) u with boundary conditions, are discretized into generalized matrix eigenvalue problems using spectral collocation. For Chebyshev bases on [-1,1], the operator becomes a matrix pencil (D_N^{(2)} + Q) \mathbf{y} = \lambda W \mathbf{y}, where D_N^{(2)} is the second differentiation matrix at Gauss-Lobatto points, Q is a diagonal matrix for q(x), and W for the weight w(x); boundary conditions modify the matrix to enforce them exactly. This yields eigenvalues and eigenfunctions with exponential convergence, often achieving relative errors below $10^{-10} for N=50 in smooth cases. A classic example is the quantum anharmonic oscillator -y'' + x^4 y = \lambda y on a truncated domain with Dirichlet boundaries, where spectral methods approximate the lowest eigenvalues with errors O(10^{-4}) for N=20, outperforming finite difference methods due to global basis resolution of oscillatory modes.^[21]^[19] For time-dependent linear PDEs, such as the heat equation u_t = \nu u_{xx}, the method of lines applies spectral discretization in space to reduce the problem to a semi-discrete system of ODEs: \frac{d\mathbf{U}}{dt} = A \mathbf{U}, where A = \nu D^2 incorporates boundary conditions. Time integration then uses implicit schemes like Crank-Nicolson, which is second-order accurate and unconditionally stable for linear problems: \mathbf{U}^{n+1} - \mathbf{U}^n = \frac{\Delta t}{2} A (\mathbf{U}^{n+1} + \mathbf{U}^n), solved via (I - \frac{\Delta t}{2} A) \mathbf{U}^{n+1} = (I + \frac{\Delta t}{2} A) \mathbf{U}^n. This approach preserves spectral spatial accuracy while allowing large time steps, with global errors O(\Delta t^2 + e^{-cN}) for smooth solutions.^[22]^[19] Parallelization of these solvers exploits the structure of spectral methods for scalability on distributed-memory systems. Domain decomposition partitions the computational domain into subdomains, solving local spectral expansions independently before global communication for boundary coupling or transforms. For Fourier-based methods, distributed FFT algorithms, such as those using 2D decomposition (transposing data across processors), enable efficient parallel computation of global transforms, achieving near-linear speedup up to thousands of cores with communication costs O(N \log N / P) for P processors. Frameworks like Dedalus implement this via tensor-product bases and MPI for multi-dimensional problems, supporting linear solvers with minimal overhead. As of 2025, GPU-accelerated packages such as cuPSS extend these capabilities to stochastic PDEs, offering higher numerical stability and performance on modern hardware.^[23]^[24]^[25]

Handling nonlinear problems

In spectral methods, nonlinear terms pose a significant challenge due to their convolution form in the spectral domain, which is computationally expensive. The pseudospectral approach addresses this by evaluating nonlinear interactions in physical space, where they reduce to simple pointwise multiplications, before projecting the result back to the spectral domain via an inverse transform. For instance, a term such as u \frac{\partial u}{\partial x} is computed by transforming the spectral coefficients to grid-point values, performing the multiplication and differentiation at collocation points (often Gaussian quadrature nodes for polynomials or evenly spaced points for Fourier bases), and then applying the forward transform to obtain the spectral representation of the nonlinearity. This strategy leverages fast Fourier transforms (FFTs) or similar discrete transforms for efficiency, achieving \mathcal{O}(N \log N) cost per evaluation, where N is the number of modes, while mitigating aliasing errors through techniques like the two-thirds rule, which truncates the convolution to avoid high-wavenumber contamination. Recent developments, such as neural spectral methods, integrate self-supervised learning in the spectral domain to enhance handling of parametric nonlinear PDEs, improving generalization as of 2024.^[10]^[1]^[26] Operator splitting techniques further facilitate the treatment of nonlinear problems by decomposing the governing equations into linear and nonlinear components, allowing each to be handled with methods suited to their structure. Linear parts, such as diffusion or advection terms, are often integrated exactly using matrix exponentials in spectral space, exploiting the diagonal or banded nature of differentiation operators, while nonlinear terms are advanced implicitly or explicitly via pseudospectral evaluation. In simulations of the Navier-Stokes equations, for example, this splitting enables stable time-stepping by treating the viscous linear operator explicitly and the nonlinear advection implicitly, often combined with pressure projection to enforce incompressibility. High-order variants, such as the operator-integration-factor method, generate accurate approximations by composing fractional steps, ensuring consistency and stability for stiff problems.^[27]^[10] For steady-state or implicit time-discretized nonlinear problems, iterative solvers like the Newton-Raphson method are employed, linearizing the residual equations around an initial guess to achieve quadratic convergence. The Jacobian matrix is formed from Fréchet derivatives in spectral space, where differentiation is precise and the structure (e.g., diagonal for Fourier bases) aids efficient solution via preconditioned Krylov methods or direct factorization. For milder nonlinearities, Picard iteration serves as a simpler fixed-point alternative, successively substituting the latest iterate into the nonlinear terms until convergence. These methods require a good initial approximation, often obtained from lower-resolution solutions or linearizations, and are particularly effective for boundary value problems arising in fluid dynamics.^[10] Conservation properties, such as discrete energy or momentum preservation, are maintained through careful choice of quadrature rules and symmetrization of nonlinear terms. In pseudospectral formulations, exact integration at collocation points ensures that the discrete inner product mimics the continuous L^2 norm, preserving quadratic invariants like energy for inviscid flows when using skew-symmetric forms (e.g., \frac{1}{2} (u \cdot \nabla) u + \frac{1}{2} \nabla (u \otimes u)) and appropriate dealiasing. For polynomial bases, Gauss-Lobatto quadrature achieves this for polynomials up to degree $2N, while Fourier methods inherently conserve energy in the absence of boundaries due to the orthogonality of exponentials. Violations can arise from aliasing or boundary treatments, but these are mitigated by over-integration or filtered projections to uphold long-term stability in simulations.^[1]^[10]

Applications

In fluid dynamics and wave propagation

Spectral methods have been extensively applied to the simulation of incompressible Navier-Stokes equations, particularly in channel flows where Fourier expansions are used in the streamwise and spanwise directions, combined with Chebyshev polynomials in the wall-normal direction to enforce no-slip boundary conditions. A key challenge in these simulations is maintaining the divergence-free condition for the velocity field. The spectral Galerkin approach addresses this through projection techniques, such as the influence matrix method, which computes the pressure implicitly by solving a Poisson equation and projecting the velocity onto a solenoidal basis to eliminate non-physical modes. This method was introduced by Kleiser and Schumann in 1980 for treating velocity-pressure coupling in complex incompressible flows, enabling high-fidelity direct numerical simulations (DNS) of turbulent channel flows at moderate Reynolds numbers. A seminal implementation is the 1987 DNS by Kim, Moin, and Moser, which resolved turbulence statistics in fully developed channel flow at Re_τ=180 (friction Reynolds number, based on friction velocity and half-channel height), demonstrating accurate prediction of mean velocity profiles and Reynolds stresses with grid resolutions of 64×96×64 modes. In turbulence simulations, spectral methods excel in direct numerical simulation of homogeneous isotropic turbulence, leveraging the periodic domain to employ global Fourier bases that diagonalize spatial derivatives and capture the full range of scales without numerical dissipation. Orszag and Patterson's 1972 pioneering work used pseudospectral Fourier methods to simulate three-dimensional isotropic turbulence at Taylor microscale Reynolds numbers up to approximately 70, resolving up to 32³ modes and verifying the decay of kinetic energy consistent with experimental wind-tunnel data. These simulations demonstrated the method's ability to handle aliasing interactions efficiently via phase-shift techniques, paving the way for higher-resolution DNS that reveal inertial-range dynamics. For forced turbulence, spectral methods maintain statistical stationarity while accurately computing energy transfer across wavenumbers. Spectral methods are also pivotal in modeling wave propagation, including linear acoustics where the wave equation is discretized using global bases like Chebyshev polynomials or Fourier series to achieve exponential convergence for smooth solutions. For the linear acoustic wave equation, spectral collocation schemes provide spectrally accurate solutions by evaluating derivatives at collocation points, as shown in numerical resolutions using high-degree polynomials with Euler time-stepping, which preserve wave speed and minimize dispersion errors over long propagation distances. In nonlinear water waves, boundary integral formulations combined with spectral expansions on the free surface enable efficient simulation of highly nonlinear phenomena, such as steep wave propagation and breaking precursors. A high-order spectral method for these equations, developed by Dommermuth and Yue in 1987, uses Fourier series for the surface elevation and potential, achieving fourth-order accuracy and simulating waves up to steepness ka=0.4 without parametric instability. For variable bathymetry, enhanced spectral boundary integral methods incorporate depth variations via conformal mapping, accurately modeling wave shoaling and refraction.^[28] Benchmark case studies highlight the precision of spectral methods in fluid problems. The Taylor-Green vortex decay serves as a canonical test for incompressible flows, initializing a smooth vortical structure whose evolution follows exact analytic expressions at low Reynolds numbers and transitions to turbulence at higher Re. Spectral simulations accurately capture the kinetic energy decay, with pseudospectral Fourier methods resolving the enstrophy growth and vortex stretching up to Re=1600, matching reference data within 1% error in integrated energy. Similarly, the Kolmogorov flow—a periodically forced shear layer—probes inverse and forward energy cascades in two- and three-dimensional turbulence using spectral methods. In 3D simulations, Fourier pseudospectral solvers reproduce the Kolmogorov -5/3 energy spectrum E(k) ~ k^{-5/3} in the inertial range for forcing wavenumbers around k_f=4, with convergence achieved at 512²×256 resolutions, validating the method's fidelity for multiscale dynamics. These cases often handle nonlinearity via techniques like pseudospectral truncation, as detailed in dedicated implementations.^[29]

In quantum mechanics and other fields

Spectral methods are widely applied in quantum mechanics to solve the time-independent Schrödinger equation for bound states, particularly through spectral collocation techniques that expand the wave function in appropriate basis sets. For the hydrogen atom, the radial part of the equation is effectively handled using Laguerre polynomials as the basis functions on a semi-infinite domain, enabling accurate computation of energy eigenvalues and eigenfunctions with high convergence rates even for excited states. This approach leverages the orthogonality of generalized Laguerre functions to discretize the differential operator, transforming the problem into a matrix eigenvalue equation solved via standard linear algebra routines.^[30]^[31] In quantum dynamics, spectral methods facilitate the simulation of time evolution by combining spatial spectral representations with split-operator techniques based on Trotter splitting. The time-dependent Schrödinger equation is propagated by alternately applying kinetic and potential energy operators in their diagonal bases—typically Fourier for momentum space and position space—achieving exponential accuracy in space and controlled error in time for wave packet dynamics in atomic and molecular systems. This method, originally developed for Cartesian coordinates and extended to spherical geometries, has become a cornerstone for studying quantum scattering and reactive processes.^[32] Beyond quantum mechanics, spectral methods address heat transfer problems in irregular geometries using Chebyshev collocation, where basis functions are projected onto non-uniform grids to handle curved boundaries and participating media. For instance, in transient conduction-radiation coupled problems within complex enclosures, Chebyshev polynomials provide spectral accuracy for temperature distributions, outperforming finite volume methods in resolution for smooth solutions. In structural vibrations, modal analysis employs spectral decomposition to extract natural frequencies and mode shapes from the eigenvalue problem of the discretized dynamic stiffness matrix, often using Fourier or polynomial bases for beam and plate structures to predict resonance behaviors efficiently.^[33]^[34]^[35] Climate modeling utilizes spherical harmonics as the spectral basis for global circulation simulations, transforming atmospheric variables from latitude-longitude grids to spectral coefficients for efficient handling of planetary-scale dynamics. This transform method, integral to models like the ECMWF Integrated Forecasting System, enables accurate representation of geostrophic flows and wave propagation on the sphere with reduced numerical diffusion compared to grid-based approaches.^[36]^[37] As of 2025, emerging applications integrate spectral methods with machine learning through spectral neural operators, which learn mappings between function spaces to approximate solutions of parametric partial differential equations without traditional meshes. These operators, grounded in Fourier or Chebyshev expansions, accelerate PDE solvers in multiphysics simulations by capturing global spectral features in data-driven ways, as demonstrated in benchmarks for diffusion and advection problems.^[38]^[39]

Comparisons with other numerical methods

Relation to finite difference and finite element methods

Spectral methods differ fundamentally from finite difference methods in their approximation strategy, employing global basis functions such as Fourier or Chebyshev polynomials that span the entire computational domain, resulting in global stencils where each degree of freedom influences all others.^[40] In contrast, finite difference methods use local stencils, approximating derivatives at a point using values from a small neighborhood, typically 3 to 5 points.^[40] This global nature enables spectral methods to capture smooth solutions with exponential convergence rates, often achieving machine precision with modest numbers of modes, whereas finite difference methods exhibit only polynomial convergence, such as second-order O(h^2) accuracy.^[40] However, finite difference methods are more robust for problems with shocks or discontinuities, as spectral methods suffer from the Gibbs phenomenon, producing oscillations near discontinuities, while finite differences can incorporate artificial dissipation to dampen such effects and perform better in non-smooth flows.^[40] Compared to finite element methods (FEM), spectral methods share a common foundation in the Galerkin projection framework, where the solution is sought in a finite-dimensional subspace, but diverge in basis selection and domain decomposition. Finite element methods typically use low-order piecewise polynomials on local elements with h-refinement (reducing element size) and p-refinement (increasing polynomial degree per element) to achieve convergence, allowing adaptive mesh refinement for complex geometries.^[41] Spectral methods, however, rely on high-order global or modal bases for p-type refinement, often requiring fewer global degrees of freedom to attain comparable accuracy due to their superior conditioning for smooth problems. While FEM distributes degrees of freedom locally across elements, leading to sparse matrices, spectral methods yield denser systems but excel in uniform, smooth domains where global modal expansions minimize error.^[41] Hybrid approaches, such as spectral volume methods, blend concepts from spectral techniques with finite volume frameworks to leverage high-order accuracy on unstructured grids while maintaining conservation properties akin to FEM.^[42] These methods subdivide control volumes into spectral subcells, reconstructing solutions via polynomial bases within each, thus combining the global accuracy of spectral methods with the geometric flexibility of finite volume discretizations.^[42] In terms of computational cost, spectral methods generally require O(N^2) operations per time step for differentiation and matrix operations in non-periodic cases, compared to O(N) for finite difference methods on N grid points, though fast transforms like FFT can reduce this to O(N log N) for periodic problems.^[22] Despite the higher per-step expense, spectral methods offer superior efficiency for high-accuracy simulations of smooth problems, as fewer points are needed to reach a given error tolerance than in finite difference or finite element approaches.^[40]

Connection to spectral element methods

The spectral element method is a high-order numerical technique that decomposes the computational domain into a mesh of elements, where each element employs local polynomial bases—such as modal or nodal expansions—to approximate the solution, with global assembly performed in a manner analogous to finite element methods while achieving spectral accuracy within each element.^[43]^[44] This approach extends pure spectral methods by retaining their high-order polynomial bases for rapid convergence but localizing the expansions to individual elements, thereby accommodating non-simply connected or complex domains that global spectral methods struggle with due to their reliance on entire-domain basis functions.^[43] Introduced by Anthony T. Patera in 1984 for fluid dynamics applications, it bridges the geometric flexibility of finite elements with spectral precision.^[44] Key features include the use of Gauss-Lobatto-Legendre quadrature points for nodal bases, ensuring C⁰ continuity across element interfaces, and support for hp-adaptive refinement, where element sizes (h) and polynomial degrees (p) are dynamically adjusted for efficiency.^[43] These methods are implemented in codes like Nekton and its successor Nek5000, widely used for computational fluid dynamics simulations involving intricate geometries.^[45]^[46] Compared to global spectral methods, spectral element approaches offer greater flexibility for handling irregular boundaries and parallelization through domain decomposition, though they require a higher number of degrees of freedom due to the multi-element structure; convergence is achieved as the aggregate of exponential rates within each element, scaling with polynomial order.^[43]

Advantages and limitations

Computational benefits

Spectral methods achieve exceptionally high accuracy for smooth solutions due to their exponential convergence rates. Unlike local methods such as finite differences or finite elements, which exhibit algebraic convergence of order O(1/N^p) for a p-th order scheme where N is the number of degrees of freedom, spectral methods yield errors that decay exponentially as O(e^{-c N}) for some constant c > 0, provided the solution is sufficiently smooth. This property arises from the global nature of the basis functions, such as Fourier or Chebyshev polynomials, which allow coefficients to decay rapidly for analytic functions. For instance, in solving singular differential equations or flows like the driven cavity problem, a modest grid of $30 \times 30 points can attain machine precision accuracy of 1 part in $10^{10}.^[10] The efficiency of spectral methods stems from their low degrees of freedom requirement to reach a desired precision \epsilon. To achieve an error below \epsilon, only N \sim \log(1/\epsilon) modes are typically needed, contrasting with the N \sim 1/\epsilon^{1/p} scaling of local methods. This is accelerated by the fast Fourier transform (FFT), which computes spectral derivatives and transforms at O(N \log N) cost, rather than O(N^2) for direct matrix operations. In practice, this enables dramatic speedups; for example, a turbulence simulation on a $128^3 grid realized a 10,000-fold performance gain over non-FFT approaches. Matrix-free implementations further enhance efficiency by avoiding explicit storage of differentiation matrices.^[10] Spectral methods exhibit excellent parallel scalability, particularly through embarrassingly parallel components like FFT-based transforms and physics computations in grid-point space. Legendre transforms and FFTs can be distributed across processors with minimal communication, achieving efficiencies up to 90% for global climate models at resolutions like T47 to T300 on distributed-memory systems. This scalability supports large-scale simulations without prohibitive overhead, as data transfers between spectral and physical spaces dominate but remain manageable.^[47] In resolving fine scales, spectral methods excel for multiscale phenomena such as turbulence cascades, where global basis functions uniformly distribute errors and efficiently capture narrow features like boundary layers or fronts. Techniques like the Witch-of-Agnesi mapping allow resolution of structures with width $2\epsilon using N \approx (d/\epsilon) \log(10) modes, enabling accurate simulation of high-Reynolds-number flows with fewer resources than local methods. This makes them particularly suited for direct numerical simulations of isotropic turbulence.^[10]

Challenges and suitability criteria

Spectral methods exhibit poor performance when applied to non-smooth solutions, such as those containing discontinuities or sharp gradients, due to the Gibbs phenomenon, which manifests as spurious oscillations near discontinuities in Fourier-based expansions and leads to a reduction in convergence rates to first-order accuracy.^[48] This limitation arises because the global nature of the basis functions assumes smoothness across the entire domain, causing slow algebraic convergence rather than the exponential rates typical for smooth problems.^[49] Additionally, handling complex geometries poses significant challenges without coordinate transformations or mappings, as the methods rely on simple domains like periodic or rectangular intervals to construct appropriate orthonormal bases effectively.^[50] Instability issues further complicate the application of spectral methods, particularly the Runge phenomenon, which occurs when using equispaced collocation points near boundaries, resulting in large oscillations and divergence for high-degree polynomial approximations.^[15] To mitigate this, Chebyshev points are commonly employed, but challenges persist with the ill-conditioning of differentiation matrices at large modal degrees N, where condition numbers grow as O(N^2), amplifying round-off errors and limiting scalability in pseudospectral implementations.^[51] These matrices are dense, exacerbating computational instability for high-resolution simulations.^[52] Spectral methods are ideally suited for problems featuring smooth, periodic solutions in simple domains, such as linear PDEs on intervals or circles, where their global basis enables high accuracy with fewer degrees of freedom compared to local methods.^[50] However, they are unsuitable for problems involving shocks, discontinuities, or irregular geometries, as these lead to aliasing and poor resolution; in such cases, hybrid approaches incorporating shock-capturing techniques are recommended over pure spectral discretizations.^[48] The choice of basis functions, such as Fourier for periodic boundaries or Chebyshev for non-periodic, influences suitability by aligning with the problem's regularity.^[15] Mitigation strategies for these challenges include spectral filtering of high-frequency modes to dampen oscillations from the Gibbs phenomenon, restoring near-exponential convergence in smooth regions while stabilizing the solution without fully eliminating the effect.^[48] For enhanced robustness, particularly with non-smooth features or complex domains, spectral methods can be combined with discontinuous Galerkin frameworks, which allow local discontinuities and better handle shocks through element-wise approximations.^[49] These hybrids leverage the high-order accuracy of spectral bases within elements while addressing global limitations.

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