A fixed grid, shifted stencil scheme for inviscid fluid-particle interaction
John D. Towers
Abstract: This paper presents a finite volume scheme for the scalar one-dimensional fluid-particle interaction model proposed in [F. Lagoutière, N. Seguin, T. Takahashi. A simple 1D model of inviscid fluid-solid interaction. J. Differential Equations, 245: 3503–3544, 2008; MR2460032]. When devising a finite volume scheme for this model, one difficulty that arises is how to deal with the moving source term in the PDE while maintaining a fixed grid. The fixed grid requirement comes from the ultimate goal of accommodating two or more particles. The finite volume scheme that we propose addresses the moving source term in a novel way. We use a modified computational stencil, with the lower part of the stencil shifted during those time steps when the particle crosses a mesh point. We then employ an altered convective flux to compensate the stencil shifts. The resulting scheme uses a fixed grid, preserves total momentum, and enforces several stability properties in the single-particle case. The single-particle scheme is easily extended to multiple particles by a splitting method.