A compatibly differenced total energy conserving form of SPH

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.

💡 Research Summary

The paper introduces a reformulation of Smoothed Particle Hydrodynamics (SPH) that guarantees exact conservation of the total energy by employing a “compatible differencing” approach for the thermal energy equation. In conventional SPH the mass and momentum equations are discretized in a fully symmetric manner, which ensures exact conservation of those quantities. However, the internal‑energy (or thermal‑energy) equation is usually written in an asymmetric form that couples pressure gradients to particle velocities through the kernel gradient. This formulation does not guarantee that the work done by inter‑particle forces is transferred precisely into internal energy, leading to systematic energy drift, especially in problems involving strong shocks.

The authors resolve this issue by redefining the internal‑energy update so that it is directly derived from the pairwise work performed by the force between two particles. For a particle pair (i, j) the force (\mathbf{F}{ij}) is computed exactly as in the momentum equation, and the work contribution is taken as (\frac{1}{2}\mathbf{F}{ij}\cdot\mathbf{v}{ij}), where (\mathbf{v}{ij}) is the relative velocity. The factor of one‑half distributes the work equally to the two interacting particles, preserving the antisymmetry of the force and guaranteeing that the sum of kinetic and internal energies is unchanged to machine precision. In compact form the compatible internal‑energy equation becomes

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