High Order Godunov Type Multimesh Method for 3d Impact Problems of Elastoplastic Media

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies under large displacements and deformations based on the multi mesh sharp interface method and modified Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The solutions of the Riemann problem for first and second order accuracy for compact stencil for an elastic medium in the case of an arbitrary stress state are obtained and presented, which are used at the “predictor” step of the Godunov scheme. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact stencil for moving Eulerian-Lagrangian grids. Modification is performed by converging the areas of influence of the differential and difference problems for the Riemann’s solver. The “corrector” step remains unchanged for both the first and second order accuracy schemes. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can change. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian-Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing obtain the parameters on the boundary and contact surfaces. The values of the underdetermined parameters in cell’s centers near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions by the Eulerian-Lagrangian and Lagrangian methods, as well as with experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter

In this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [29]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. The modification in this paper improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter.

A Novel High-Order Accurate Compact Stencil Poisson Solver: Application to Cavity Flows

In this paper, a fourth-order compact stencil finite difference scheme is developed for solving elliptic Poisson equation. The scheme presented here is based on a modular approach using a linear combination of compact difference algorithms that results in different discrete formulation than the well-known Mehrstellen scheme. An adjoint optimal V-cycle multigrid (MG) iterative solver are developed, implemented, and tested. The robustness of the adjoint Poisson solver is illustrated by solving incompressible Navier–Stokes equations in vorticity-stream function formulation. Using a fully implicit factorized delta-scheme algorithm for the time integration, benchmark quality results of the cavity flow problem are presented and compared to existing literature for various Reynolds numbers.

An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators

We develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes.

A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations

In this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values ofu,(∂2u/∂n2) and (∂4u/∂n4) are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

A Novel Numerical Method of for Three-Dimensional Non-Linear Triharmonic Equations

In this article, we present two new novel finite difference approximations of order two and four, respectively, for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u, ∂2u/∂n2 and ∂4u/∂n4 are prescribed on the boundary. We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions. We require only 7- and 19-grid points on the compact cell for the second and fourth order approximation, respectively. The Laplacian and the biharmonic of the solution are obtained as by-product of the methods. We require only system of three equations to obtain the solution. Numerical results are provided to illustrate the usefulness of the proposed methods.