Higher order semi-implicit schemes for linear advection equation on Cartesian grids with numerical stability analysis

TL;DR

This paper introduces semi-implicit schemes inspired by $$-schemes for the linear advection equation, achieving unconditional stability in 1D and second order accuracy in higher dimensions, with numerical stability analysis and experiments.

Contribution

It develops a new semi-implicit scheme based on $$-schemes that is unconditionally stable and second order accurate in multiple dimensions, extending explicit methods.

Findings

Unconditionally stable in 1D for the semi-implicit scheme.

Second order accuracy in higher dimensions.

Numerical experiments confirm stability and accuracy.

Abstract

A new class of semi-implicit numerical schemes for linear advection equation on Cartesian grids is derived that is inspired by so-called $\kappa$-schemes used with fully explicit discretizations for this type of problems. Opposite to fully explicit $\kappa$-scheme the semi-implicit variant is unconditionally stable in one-dimensional case and it preserves second order accuracy for dimension by dimension extension in higher dimensional cases. We discuss von Neumann stability conditions numerically for all numerical schemes. Using so-called Corner Transport Upwind extension of two-dimensional semi-implicit scheme with a special choice of $\kappa$ parameters, a second order accurate method is obtained for which numerical unconditional stability can be shown for variable velocity and the third order accuracy can be proved for constant velocity. Several numerical experiments illustrate the properties of semi-implicit schemes for chosen examples.