Semi-implicit methods for advection equations with explicit forms of numerical solution

TL;DR

This paper introduces a family of semi-implicit, second-order accurate numerical methods for advection equations that are unconditionally stable and applicable to variable velocity scenarios, using a novel combination of implicit-explicit discretizations.

Contribution

The paper proposes a new class of semi-implicit methods with fixed iteration counts and variable parameters, enhancing stability and accuracy for advection equations with variable velocities.

Findings

Methods are unconditionally stable for variable velocity.

Numerical experiments demonstrate improved accuracy and stability.

Differential programming optimizes variable numerical parameters.

Abstract

We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward alternating substitutions. The methods use a novel combination of implicit and explicit time discretizations for one-dimensional case and the Strang splitting method in several dimensional case. The methods are described for advection equations with a continuous variable velocity that can change its sign inside of computational domain. The methods are unconditionally stable in the non-conservative case for variable velocity and for variable numerical parameter. Several numerical experiments confirm the advantages of presented methods including an involvement of differential programming to find optimized values of the variable numerical parameter.