A computation with tree-based AMR method using multi-moment scheme for conservative phase-field equation with a flux term

A phase-field model describes the gas-liquid interface of two-phase flows. We solve the conservative Allen-Cahn equation combined with the continuum equation on a two-dimensional computational domain with a given velocity field. The accuracy of numerical result strongly depends on the mesh resolution around the interface. The AMR (Adaptive Mesh Refinement) method greatly reduces the computational cost, since it is possible to assign high-resolution mesh to the region around the moving interface. We have developed a code to solve the equation in a manner of the tree-based AMR with multi-moment methods, the conservative IDO and CIP-CSL schemes. To reduce the implementation difficulties of AMR method, we introduce the fractional step method and the directional splitting method. In a benchmark test of the single vortex problem, the AMR computation with 5-level refinement for the interface achieves 9.26-times speed up and 1/12.3 mesh reduction to compare with the computation on a uniform mesh.