Analysis of two fully discrete spectral volume schemes for hyperbolic equations
Recommended citation: Wei P, Zou Q. Analysis of two fully discrete spectral volume schemes for hyperbolic equations[J]. Numerical Methods for Partial Differential Equations, 2024, 40(2): e23072. https://onlinelibrary.wiley.com/doi/abs/10.1002/num.23072
- First published: 20 September 2023
Abstract
In this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one-dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second-order Runge-Kutta (RK2) method in time-discretization, and by letting a piecewise $k$th degree ($k \geq 1$ is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with $k$ Gauss-Legendre points (LSV) or right-Radau points (RRSV). We prove that for the EU-SV schemes, the weak(2) stability holds and the $L_{2}$ norm errors converge with optimal orders $\mathscr{O}\left(h^{k+1}+\tau\right)$, provided that the CFL condition $\tau \leq C h^{2}$ is satisfied. While for the RK2-SV schemes, the weak(4) stability holds and the $L_{2}$ norm errors converge with optimal orders $\mathscr{O}\left(h^{k+1}+\tau^{2}\right)$, provided that the CFL condition $\tau \leq C h^{\frac{4}{3}}$ is satisfied. Here $h$ and $\tau$ are, respectively, the spacial and temporal mesh sizes and the constant $C$ is independent of $h$ and $\tau$. Our theoretical findings have been justified by several numerical experiments.
Recommended Citation
- Wei P, Zou Q. Analysis of two fully discrete spectral volume schemes for hyperbolic equations[J]. Numerical Methods for Partial Differential Equations, 2024, 40(2): e23072.
BibTeX
@article{wei2024analysis,
title={Analysis of two fully discrete spectral volume schemes for hyperbolic equations},
author={Wei, Ping and Zou, Qingsong},
journal={Numerical Methods for Partial Differential Equations},
volume={40},
number={2},
pages={e23072},
year={2024},
publisher={Wiley Online Library}
}