How to train your solver: A method of manufactured solutions for weakly-compressible SPH

TL;DR

This paper introduces the method of manufactured solutions (MMS) for systematically testing and verifying the accuracy, convergence, and boundary conditions of weakly-compressible SPH solvers, including recent second order schemes.

Contribution

It adapts the MMS approach for Lagrangian WCSPH solvers, enabling comprehensive and efficient verification in 2D and 3D, which was not previously established.

Findings

MMS effectively tests convergence and accuracy of WCSPH schemes.

The method identifies solver issues and boundary condition problems.

Open source implementation ensures reproducibility.

Abstract

The Weakly-Compressible Smoothed Particle Hydrodynamics (WCSPH) method is a Lagrangian method that is typically used for the simulation of incompressible fluids. While developing an SPH-based scheme or solver, researchers often verify their code with exact solutions, solutions from other numerical techniques, or experimental data. This typically requires a significant amount of computational effort and does not test the full capabilities of the solver. Furthermore, often this does not yield insights on the convergence of the solver. In this paper we introduce the method of manufactured solutions (MMS) to comprehensively test a WCSPH-based solver in a robust and efficient manner. The MMS is well established in the context of mesh-based numerical solvers. We show how the method can be applied in the context of Lagrangian WCSPH solvers to test the convergence and accuracy of the solver in two and three dimensions, systematically identify any problems with the solver, and test the boundary conditions in an efficient way. We demonstrate this for both a traditional WCSPH scheme as well as for some recently proposed second order convergent WCSPH schemes. Our code is open source and the results of the manuscript are reproducible.