Abstract
The first-order Lax-Friedrichs (LF) scheme is commonly used in conjunction with other schemes to achieve monotone and stable properties with lower numerical diffusion. Nevertheless, the LF scheme and the schemes devised based on it, for example, the first-order centered (FORCE) and the slope-limited centered (SLIC) schemes, cannot achieve a time-step-independence solution due to the excessive numerical diffusion at a small time step. In this work, two time-step-convergence improved schemes, the C-FORCE and C-SLIC schemes, are proposed to resolve this problem. The performance of the proposed schemes is validated in solving the one-layer and two-layer shallow-water equations, verifying their capabilities in attaining time-step-independence solutions and showing robustness of them in resolving discontinuities with high-resolution.
Highlights
The Lax-Friedrichs (LF) scheme, called the Lax method [1], is a classical explicit three-point scheme in solving partial differential equations in, for example, aerodynamics, hydrodynamics, and magnetohydrodynamics [2,3,4]
When solving the linear convection problem, compared with the LF scheme, the numerical viscosity of the first-order centered (FORCE) scheme is reduced by half [4]
The LF scheme and the schemes devised based on it have a common shortcoming, which has raised less attention so far. We demonstrate this shortcoming by using the LF scheme to solve the following linear convection problem in the timespace domain [0, T] × [0, L]: PDE:
Summary
The Lax-Friedrichs (LF) scheme, called the Lax method [1], is a classical explicit three-point scheme in solving partial differential equations in, for example, aerodynamics, hydrodynamics, and magnetohydrodynamics [2,3,4]. When solving the linear convection problem, compared with the LF scheme, the numerical viscosity of the FORCE scheme is reduced by half [4]. In some situations, for instance, when rapid wet-dry transitions exist (e.g., for simulations in the nearshore regions), or when some other constraints on the time step (e.g., those due to surface tension and fluid viscosity effects [10]) dominate the CFL condition, one may have to use a rather small time step compared with that determined by the CFL condition In these situations, the LF-based schemes are inappropriate due to the excessive numerical diffusion at a small value of Δt.
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