Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

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-or...

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Bibliographic Details
Published inAxioms Vol. 11; no. 1; p. 28
Main Authors Takei, Yasuhiro, Iwata, Yoritaka
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.01.2022
Subjects
Approximation
Boundary conditions
Conservation laws
Discretization
Evolution
Fourier spectral method
Fourier transforms
high-precision numerical scheme
implicit Runge-Kutta method
Iterative methods
Mathematical analysis
Methods
Partial differential equations
Runge-Kutta method
Spectral methods
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Summary: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.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms11010028