High-resolution Algorithms Based On Adaptive Moving Mesh

The numerical solutions of hyperbolic conservation laws are always one of cores on the computational fluid dynamics.For the nonlinear hyperbolic conservation laws equations,the solution may appear discontinuity at some time,even if the initial value is sufficiently smooth.When the uniform grids is used for the numerical computation,usually the number of the nodes is nice for the big change solution but is too much for the little change solution,or the number of the nodes is enough for the little change solution while it is less for the big change solution.In a word,it will cause a huge waste of computational resources when the many uniform grids are utilized for the problem with little change solution.And it will not obtain accurate enough solutions when the less uniform grids are utilized for the problem with the big change solution.So,the reasonable grid distribution plays an important role for the efficient and the accurate computation of the nonlinear hyperbolic conservation laws equations.Furthermore,a more universal moving mesh algorithm is constructed by the new monitor function and a conservative upwind scheme on the basis of the moving mesh algorithm.And the new schemes are used to solve the classic examples of the Burgers equation and the 1D and 2D Euler equations.The results show the performance of the new scheme.The details in this paper are as follows:(1)The high-resolution entropy stable scheme based on the moving mesh algorithm is constructed for one-dimensional hyperbolic conservation laws equations.The new monitor function can capture not only the contact wave and the shock wave but also the rarefaction wave which can effectively improve the serious smearing problem for the rarefaction wave.Moreover,a conservative upwind scheme is constructed to reduce the accumulation of the truncation error and obtain the physical quantity at the new grids for less error.And the high resolution entropy stable schemes is used to discretized the equations.The numerical results are obtained by the new schemes for the 1D Burgers equation and Euler equation,comparing with the numerical results which are obtained by exact solution and the existing algorithm(high-resolution schemes based on structure mesh),the numerical results phenomena(e.g.glitch wave,over-shooting etc.).(2)For the two-dimensional hyperbolic conservation laws equations,in space,the moving mesh algorithm is used for the mesh redistribution,which leads the mesh in computational domain is discretized into unstructured grid.Therefore,the entropy stable scheme is constructed based on unstructured grid.For two dimensional Euler equations,the entropy-stable scheme based on moving mesh algorithm is constructed and the results of numerical simulations compare the numerical results of the new algorithm and the other schemes,which verify that the new scheme not only has high resolution,but also can sharply capture shock waves and rarefaction.(3)The entropy-stable scheme has its advantages for the numerical solutions of hyperbolic conservation laws,but it needs to obtain the entropy function and entropy flux function for different equations to deduce the specific entropy stable scheme,which make the scheme poor universal.And then,the Kinetic Flux Vector Splitting scheme based on the adaptive moving mesh algorithm is constructed for two dimensional Euler equations.The advantages of new scheme can be verified by the results of numerical simulations are compared with the numerical results of the other schemes.