Abstract
The paper constructs a class of simple high-accurate schemes (SHA schemes) with third order approximation accuracy in both space and time to solve linear hyperbolic equations, using linear data reconstruction and Lax-Wendroff scheme. The schemes can be made even fourth order accurate with special choice of parameter. In order to avoid spurious oscillations in the vicinity of strong gradients, we make the SHA schemes total variation diminishing ones (TVD schemes for short) by setting flux limiter in their numerical fluxes and then extend these schemes to solve nonlinear Burgers’ equation and Euler equations. The numerical examples show that these schemes give high order of accuracy and high resolution results. The advantages of these schemes are their simplicity and high order of accuracy.
Highlights
In designing numerical schemes of very high order of accuracy for solving hyperbolic conservation laws one faces at least three major difficulties
The other very important issue relates to the generation of spurious oscillations in the vicinity of strong gradients
State-of-the-art very high order methods for hyperbolic conservation laws include the class of ENO/WENO schemes [2,3,4], Spectral Method [5], the class of Compact Difference Methods [6], Discontinuous Galerkin Finite Element Methods [7], and ADER Methods [8]
Summary
In designing numerical schemes of very high order of accuracy for solving hyperbolic conservation laws one faces at least three major difficulties. As regards the ENO/WENO/MPWENO approach, the most accurate scheme reported so far uses 9th order spatial discretisation with Runge-Kutta methods for time evolution. One of them is the MUSCL (Monotone Upstream-Centred Scheme for Conservation Laws) [9,10,11] approach which is a very popular and simple one to construct explicit, fully discrete schemes that lies on the solution of Riemann problem and could be extended. We consider the Initial Boundary Value Problem (IBVP) for the linear advection equation in the domain [0, L]×[0, T] on the x−t plane.
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