2022
DOI: 10.3390/axioms11010028
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Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

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 di… Show more

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Cited by 6 publications
(4 citation statements)
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“…On the other hand, they have disadvantages such as high computational cost compared to conventional methods. The advent of spectral methods has made it possible to combine high accuracy with low computational cost [1,38,55], they are methods characterized by exponential convergence to the exact solution with increasing grid size.…”
Section: Introductionmentioning
confidence: 99%
“…On the other hand, they have disadvantages such as high computational cost compared to conventional methods. The advent of spectral methods has made it possible to combine high accuracy with low computational cost [1,38,55], they are methods characterized by exponential convergence to the exact solution with increasing grid size.…”
Section: Introductionmentioning
confidence: 99%
“…Moreover, they have effective error estimators, which can provide an estimate of the local or global error of the numerical solution. These features make IRK methods suitable for optimizing the time step needed to ensure stable and accurate solutions while maintaining the dispersion and dissipation at fixed levels [19][20][21]. Implicit Runge-Kutta (IRK) methods are a cutting-edge group of schemes within the range of advanced time discretization schemes [22].…”
Section: Introductionmentioning
confidence: 99%
“…Because of the aforementioned applications, various numerical techniques for solving various application problems, such as time-frequency analysis, signal delay, convection-diffusion equations, nonlinear approximation, and Monte Carlo simulation, arising in fluid dynamics problems have been developed. For instance, predictor-corrector techniques (Su et al [11], Awoyemi and Idowu [12], Iskandarov and Komartsova [13], Ashry et al [14], Asif [15]); Galerkin methods (Guo et al [16]); Haar wavelets methods (Aziz and Khan [17], Shiralashetti et al [18], Saparova et al [19]); Runge-Kutta methods (Takei and Iwata [20], Yakubu et al [21], Zhao and Huang [22]); Newtral network model (Mall and Chakraverty [23]); multigrid technique (Ghaffar et al [24], Ge [25], Gupta et al [26]); finite difference methods (Mulla et al [27]); and finite element methods (Harari and Hughes [28]).…”
Section: Introductionmentioning
confidence: 99%