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Register a new userElastic wave propagation in curvilinear coordinates with mesh refinement interfaces by a fourth order finite difference method
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Added by
Siyang Wang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the material property can be discontinuous at curved interfaces. The governing equations are discretized in second order form on curvilinear meshes by using a fourth order finite difference operator satisfying a summation-by-parts property. The method is energy stable and high order accurate. The highlight is that mesh sizes can be chosen according to the velocity structure of the material so that computational efficiency is improved. At the mesh refinement interfaces with hanging nodes, physical interface conditions are imposed by using ghost points and interpolation. With a fourth order predictor-corrector time integrator, the fully discrete scheme is energy conserving. Numerical experiments are presented to verify the fourth order convergence rate and the energy conserving property.
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We develop a stable finite difference method for the elastic wave equations in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equations are discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In numerical experiments, we demonstrate that the convergence rate is optimal, and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as given by the usual Courant Friedrichs Lewy condition.
Discrete approximations to the equation begin{equation*} L_{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A(x)+H(x)) u^{(1)} + B(x) u = f, ; xin[0,1] end{equation*} are considered. This is an extension of the Sturm-Liouville case $D(x)equiv H(x)equiv 0$ [ M. Ben-Artzi, J.-P. Croisille, D. Fishelov and R. Katzir, Discrete fourth-order Sturm-Liouville problems, IMA J. Numer. Anal. {bf 38} (2018), 1485-1522. doi: 10.1093/imanum/drx038] to the non-self-adjoint setting. The natural boundary conditions in the Sturm-Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second, third and fourth order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties--compactness and coercivity. It allows to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.
We develop an energy-based finite difference method for the wave equation in second order form. The spatial discretization satisfies a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The energy-based discretization does not use any mesh-dependent parameter, even in the presence of Dirichlet boundary conditions and material interfaces. Furthermore, similar to upwind discontinuous Galerkin methods, numerical dissipation can be added to the discretization through the boundary conditions. We present numerical experiments that verify convergence and robustness of the proposed method.
A high fidelity flow simulation for complex geometries for high Reynolds number ($Re$) flow is still very challenging, which requires more powerful computational capability of HPC system. However, the development of HPC with traditional CPU architecture suffers bottlenecks due to its high power consumption and technical difficulties. Heterogeneous architecture computation is raised to be a promising solution of difficulties of HPC development. GPU accelerating technology has been utilized in low order scheme CFD solvers on structured grid and high order scheme solvers on unstructured meshes. The high order finite difference methods on structured grid possess many advantages, e.g. high efficiency, robustness and low storage, however, the strong dependence among points for a high order finite difference scheme still limits its application on GPU platform. In present work, we propose a set of hardware-aware technology to optimize the efficiency of data transfer between CPU and GPU, and efficiency of communication between GPUs. An in-house multi-block structured CFD solver with high order finite difference methods on curvilinear coordinates is ported onto GPU platform, and obtain satisfying performance with speedup maximum around 2000x over a single CPU core. This work provides efficient solution to apply GPU computing in CFD simulation with certain high order finite difference methods on current GPU heterogeneous computers. The test shows that significant accelerating effects can been achieved for different GPUs.
This paper develops high-order accurate entropy stable (ES) adaptive moving mesh finite difference schemes for the two- and three-dimensional special relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which is the high-order accurate extension of [J.M. Duan and H.Z. Tang, Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics, J. Comput. Phys., 426(2021), 109949]. The key point is the derivation of the higher-order accurate entropy conservative (EC) and ES finite difference schemes in the curvilinear coordinates by carefully dealing with the discretization of the temporal and spatial metrics and the Jacobian of the coordinate transformation and constructing the high-order EC and ES fluxes with the discrete metrics. The spatial derivatives in the source terms of the symmetrizable RMHD equations and the geometric conservation laws are discretized by using the linear combinations of the corresponding second-order case to obtain high-order accuracy. Based on the proposed high-order accurate EC schemes and the high-order accurate dissipation terms built on the WENO reconstruction, the high-order accurate ES schemes are obtained for the RHD and RMHD equations in the curvilinear coordinates. The mesh iteration redistribution or adaptive moving mesh strategy is built on the minimization of the mesh adaption functional. Several numerical tests are conducted to validate the shock-capturing ability and high efficiency of our high-order accurate ES adaptive moving mesh methods on the parallel computer system with the MPI communication. The numerical results show that the high-order accurate ES adaptive moving mesh schemes outperform both their counterparts on the uniform mesh and the second-order ES adaptive moving mesh schemes.
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