Explicit and Implicit Kinetic Streamlined-Upwind Petrov Galerkin Method for Hyperbolic Partial Differential Equations

TL;DR

This paper introduces a new explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme that improves stability and accuracy for hyperbolic PDEs like Burgers and Euler equations, with comprehensive numerical validation.

Contribution

The paper develops a novel KSUPG scheme that outperforms traditional SUPG methods in multi-dimensional hyperbolic problems, including stability analysis and comparison of explicit and implicit versions.

Findings

Enhanced numerical stability and accuracy for hyperbolic PDEs.

Superior performance of KSUPG over traditional SUPG in multi-dimensions.

Spectral stability confirmed for explicit 2D formulation.

Abstract

A novel explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme is presented for hyperbolic equations such as Burgers equation and compressible Euler equations. The proposed scheme performs better than the original SUPG stabilized method in multi-dimensions. To demonstrate the numerical accuracy of the scheme, various numerical experiments have been carried out for 1D and 2D Burgers equation as well as for 1D and 2D Euler equations using Q4 and T3 elements. Furthermore, spectral stability analysis is done for the explicit 2D formulation. Finally, a comparison is made between explicit and implicit versions of the KSUPG scheme.