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.
Highlights
Initial-value problems are traditionally solved numerically, using finite steps for the temporal domain
The generalized weighted residual method (GWRM) is primarily intended for computing long time behaviour of complex problems with several time scales, it can be used for accurate solution of stiff problems
For nonlinear problems, where all terms must be advanced implicitly, the Crank-Nicolson method is expected to compare less favourably with the GWRM either due to reduced efficiency when solving a nonlinear system of equations at each iteration or due to reduced accuracy if nonlinear terms are time linearized
Summary
Initial-value problems are traditionally solved numerically, using finite steps for the temporal domain. Semiimplicit methods [2,3] allow large time steps and more efficient matrix inversions than those of implicit methods, but may feature limited accuracy. These methods provide sufficiently efficient and accurate solutions for most applications. For applications in physics where there exist several separated time scales, the numerical work relating to advancement in the time domain can become very demanding Another computational issue is that it may be advantageous to determine parametrical dependences without performing a sequence of runs for different choices of parameter values
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