A Spectral Method in Time for Initial-Value Problems

Abstract

A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

Uncertainty propagation by using spectral methods: A practical application to a two-dimensional turbulence fluid model

“…By applying the WRM, a system of nonlinear algebraic equations is derived for the coefficients of the Chebyshev expansion. The solution of these equations directly provides information on the dependence of the simulation results on the input parameters [11].…”

“…We note that the methodology illustrated in the rest of the present section is based on the work presented in Ref. [11]. We consider an initial value parabolic or hyperbolic partial differential equation…”

“…In Fig. 4 we present the distributions of η obtained at x = 40 (first row) and x = 60 (second row) with the spectral code for (K, L, M, N ) = (1, 12, 9, 1), (2,14,11,2), (3,16,13,3) (left, middle, and right columns, respectively). We observe that the distributions slightly depend on the number of spectral terms used in the simulations and that η does not depend linearly on…”

“…[11], which consists in a decomposition of the solution to the model equations in terms of Chebyshev polynomials along the time, spatial, and input parameter coordinates. More precisely, a series of Chebyshev polynomials is used to represent the solution of a differential equation and to express its dependence on the temporal, spatial, and input variables.…”

“…However, fully spectral codes were rarely used to investigate plasma physics problems and, to our knowledge, their use remained limited to the study of plasma flows and linear stability analysis (see, e.g., Refs. [11,19]), not being applied to the analysis of uncertainty propagation in nonlinear plasma turbulence simulations. This motivates the study illustrated in the present paper.…”