The Piecewise Cubic Method (PCM) for Computational Fluid Dynamics

TL;DR

The paper introduces a high-order finite volume reconstruction method called PCM for hyperbolic conservation laws, achieving fifth-order spatial accuracy and fourth-order temporal accuracy with improved efficiency over traditional methods.

Contribution

A novel piecewise cubic polynomial method that simplifies and enhances the accuracy and efficiency of high-order CFD simulations for hyperbolic laws.

Findings

Converges at fifth-order in 1D smooth flows

Demonstrates effectiveness in gas dynamics and magnetohydrodynamics

Offers computational efficiency over Runge-Kutta methods

Abstract

We present a new high-order finite volume reconstruction method for hyperbolic conservation laws. The method is based on a piecewise cubic polynomial which provides its solutions a fifth-order accuracy in space. The spatially reconstructed solutions are evolved in time with a fourth-order accuracy by tracing the characteristics of the cubic polynomials. As a result, our temporal update scheme provides a significantly simpler and computationally more efficient approach in achieving fourth order accuracy in time, relative to the comparable fourth-order Runge-Kutta method. We demonstrate that the solutions of PCM converges in fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme in a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.