Two-level schemes for the advection equation

TL;DR

This paper develops and analyzes stable two-level numerical schemes for the advection equation, ensuring properties like conservation and monotonicity, with a focus on explicit, implicit, and regularized schemes.

Contribution

It introduces new stability conditions for explicit schemes and constructs unconditionally stable implicit schemes, improving numerical solutions of the advection equation.

Findings

Explicit scheme is absolutely unstable without modifications.

Implicit schemes like Crank-Nicolson are unconditionally stable.

Numerical results confirm the accuracy and stability of the proposed schemes.

Abstract

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standart explicit two-level scheme for the advection equation is absolutly unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax-Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank-Nicolson scheme) and fourth order. The conditionally stable implicit Lax-Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.