A compatibly differenced total energy conserving form of SPH

TL;DR

This paper introduces a modified SPH method that ensures exact total energy conservation by using a compatibly differenced formalism, improving accuracy and convergence in shock tests.

Contribution

The paper presents a new energy-conserving SPH formulation based on compatible differencing, enhancing accuracy over standard methods.

Findings

The compatible SPH achieves reliable first-order convergence.

It improves accuracy in shock test problems.

It guarantees exact total energy conservation.

Abstract

We describe a modified form of Smoothed Particle Hydrodynamics (SPH) in which the specific thermal energy equation is based on a compatibly differenced formalism, guaranteeing exact conservation of the total energy. We compare the errors and convergence rates of the standard and compatible SPH formalisms on analytic test problems involving shocks. We find that the new compatible formalism reliably achieves the expected first-order convergence in such tests, and in all cases improves the accuracy of the numerical solution over the standard formalism.