The Piecewise Cubic Method (PCM) for Computational Fluid Dynamics

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.

58 pages, 16 figures

References in corpus (5)

• PLUTO: a Numerical Code for Computational Astrophysics
• Athena: A New Code for Astrophysical MHD
• An Unsplit Godunov Method for Ideal MHD via Constrained Transport in Three Dimensions
• Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free Magnetohydrodynamics
• Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

Cited by in corpus (6)

• Magnetohydrodynamic Simulation Code CANS+: Assessments and Applications
• A $4^{\mathrm{th}}$-order accurate finite volume method for ideal classical and special relativistic MHD based on pointwise reconstructions
• A high-order weighted finite difference scheme with a multi-state approximate Riemann solver for divergence-free magnetohydrodynamic simulations
• A variable high-order shock-capturing finite difference method with GP-WENO
• GP-MOOD: A positive-preserving high-order finite volume method for hyperbolic conservation laws
• A single-step third-order temporal discretization with Jacobian-free and Hessian-free formulations for finite difference methods