Abstract
A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is presented. The order of total calculation cost is O(Nlog2N). As a benchmark, the relations between numerical accuracy and discretization unit size and that between the stability of calculation and discretization unit size are demonstrated for both linear and nonlinear cases.
Highlights
The dynamics of nonlinear hyperbolic equations are fascinating enough to be applicable to wave propagation on any scale, from elementary particles to waves on a cosmic scale
RK uerr,N5 and RK uerr,N10, RK verr,N5 and RK verr,N10 denote the error of u and v, when the implicit Runge-Kutta method is applied with N = 25 and 210, respectively
Uerr,t6, RK uerr,t16, RK verr,t6 and RK verr,t16 denote the error in u and v, when the implicit Runge-Kutta method is applied with the time increments ∆t = 2−6, 2−13, respectively
Summary
The dynamics of nonlinear hyperbolic equations are fascinating enough to be applicable to wave propagation on any scale, from elementary particles to waves on a cosmic scale. While in terms of treating nonlinear problems (e.g., various types of boundary, discontinuity such as shock propagation) it is important to individually specialize numerical schemes, we are going to establish a basic framework for calculating nonlinear hyperbolic evolution equations. In this context, much attention is paid to “universal applicability” and “reliability”. While linear and nonlinear solvers based on explicit methods are relatively simple with low computation cost, it is known that there is a restriction called the Courant-Friedrichs-Lewy Condition (CFL condition, for short) on the time unit ∆t in order to obtain numerical results stably. By incorporating spectral elements methods, spectral penalty methods, and so on, it is definitely possible to construct numerical schemes for the other boundary conditions
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