The Piecewise Cubic Method (PCM) for computational fluid dynamics

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 b...

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Bibliographic Details
Published in:Journal of computational physics 2017-07, Vol.341, p.230-257
Main Authors: Lee, Dongwook, Faller, Hugues, Reyes, Adam
Format: Article
Language:English
Subjects:
Computational fluid dynamics
Computational physics
Conservation laws
Eulers equations
Finite volume method
Fluid dynamics
Gas dynamics
Gas flow
Godunov's method
High-order methods
Magnetohydrodynamics
Piecewise cubic method
Polynomials
Runge-Kutta method
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Summary: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 at fifth-order in solving 1D smooth flows described by hyperbolic conservation laws. We test the new scheme on a range of numerical experiments, including both gas dynamics and magnetohydrodynamics applications in multiple spatial dimensions.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2017.04.004