Semi-Implicit-type order-adaptive CAT2 schemes for systems of balance laws with relaxed source term

TL;DR

This paper introduces semi-implicit second order CAT2 schemes combined with MOOD detection for solving hyperbolic balance laws, achieving high accuracy, non-oscillatory behavior, and positivity preservation across diverse test cases.

Contribution

It presents a novel blend of semi-implicit CAT2 schemes with MOOD for improved accuracy and robustness in hyperbolic balance law simulations.

Findings

High accuracy on smooth solutions

Non-oscillatory behavior for irregular solutions

Effective positivity preservation

Abstract

In this paper we present two semi-implicit-type second order Compact Approximate Taylor (CAT2) numerical schemes and blend them with a local a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of balance laws with relaxed source term. The resulting scheme presents high accuracy when applied to smooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearly fail-safe property in terms of ensuring positivity. The numerical results obtained from a variety of test cases, including smooth and non-smooth well-prepared and unprepared initial condition, assessing the appropriate behavior of the semi-implicit-type second order CATMOOD schemes. These results have been compared in accuracy and efficiency with a second order semi-implicit Runge-Kutta (RK) method.