A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes

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In this paper, we present high order accurate positivity-preserving conservative remapping algorithm which is based on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. We use a third-order method on 2D quadrilateral meshes as an example to present the algorithm. The method can effectively remap the physical variables after mesh rezoning in the ALE algorithm. By calculating the intersection exactly, this method does not require the same connectivity between the old and new meshes. By reconstructing a quadratic polynomial and a zero-order polynomial for each cell in a two-dimensional domain, this method assigns nonlinear weights for these polynomials accordingly after calculating the smoothness indicators over the integration area, yielding third order accuracy without numerical oscillations. After calculating the overlaps between the old and new meshes and integrating the polynomials over the intersections, the remapping is completed. Furthermore, to ensure the positivity-preserving property of relevant physical variables in hydrodynamics numerical simulation such as density and internal energy, a simple and efficient positive-preserving limiter is adopted to slightly modify the reconstructed polynomials, which can maintain the original order of accuracy and conservation. The algorithm can be extended to higher order accuracy using higher order reconstruction and higher order integration formula over the intersection areas. A series of numerical experiments are performed to test the properties of the multi-resolution WENO conservative remapping algorithm. Numerical results show that the algorithm is conservative, positivity-preserving, highly efficient, third-order accurate for smooth problems, and essentially non-oscillatory for discontinuous problems.