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- Title
Analysis of two fully discrete spectral volume schemes for hyperbolic equations.
- Authors
Wei, Ping; Zou, Qingsong
- 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 kth degree(k≥1$$ k\ge 1 $$ is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with k$$ k $$ Gauss–Legendre points (LSV) or right‐Radau points (RRSV). We prove that for the EU‐SV schemes, the weak(2) stability holds and the L2$$ {L}_2 $$ norm errors converge with optimal orders 풪(hk+1+τ), provided that the CFL condition τ≤Ch2$$ \tau \le C{h}^2 $$ is satisfied. While for the RK2‐SV schemes, the weak(4) stability holds and the L2$$ {L}_2 $$ norm errors converge with optimal orders 풪(hk+1+τ2), provided that the CFL condition τ≤Ch43$$ \tau \le C{h}^{\frac{4}{3}} $$ is satisfied. Here h$$ h $$ and τ$$ \tau $$ are, respectively, the spacial and temporal mesh sizes and the constant C$$ C $$ is independent of h$$ h $$ and τ$$ \tau $$. Our theoretical findings have been justified by several numerical experiments.
- Subjects
EQUATIONS; RUNGE-Kutta formulas; CONSERVATION laws (Physics); EULER method; CONSERVATION laws (Mathematics); INTEGERS
- Publication
Numerical Methods for Partial Differential Equations, 2024, Vol 40, Issue 2, p1
- ISSN
0749-159X
- Publication type
Article
- DOI
10.1002/num.23072