Magnetohydrodynamic code for gravitationally-stratified media

Aims. We describe a newly-developed magnetohydrodynamic (MHD) code with the capacity to simulate the interaction of any arbitrary perturbation (i.e., not necessarily limited to the linearised limit) with a magnetohydrostatic equilibrium background. Methods. By rearranging the terms in the system of MHD equations and explicitly taking into account the magnetohydrostatic equilibrium condition, we define the equations governing the perturbations that describe the deviations from the background state of plasma for the density, internal energy and magnetic field. We found it was advantageous to use this modified form of the MHD equations for numerical simulations of physical processes taking place in a stable gravitationally-stratified plasma. The governing equations are implemented in a novel way in the code. Sub-grid diffusion and resistivity are applied to ensure numerical stability of the computed solution of the MHD equations. We apply a fourth-order central difference scheme to calculate the spatial derivatives, and implement an arbitrary Runge-Kutta scheme to advance the solution in time. Results. We have built the proposed method, suitable for strongly-stratified magnetised plasma, on the base of the well-documented Versatile Advection Code (VAC) and performed a number of one- and multi-dimensional hydrodynamic and MHD tests to demonstrate the feasibility and robustness of the code for applications to astrophysical plasmas

Orthogonal subgrid-scale stabilization for nonlinear reaction-convection-diffusion equations

Nonlinear reaction-convection-diffusion equations are encountered in modeling of a variety of natural phenomena such as in chemical reactions, population dynamics and contaminant dispersal. When the scale of convective and reactive phenomena are large, Galerkin finite element solution fails. As a remedy, Orthogonal Subgrid Scale stabilization is applied to the finite element formulation. It has its origins in the Variational Multi Scale approach. It is based on a fine grid - coarse grid component sum decomposition of solution and utilizes the fine grid solution orthogonal to the residual of the finite element coarse grid solution as a correction term. With selective mesh refinement, a stabilized oscillation-free solution that can capture sharp layers is obtained. Newton Raphson method is utilized for the linearization of nonlinear reaction terms. Backward difference scheme is used for time integration. The formulation is tested for cases with standalone and coupled systems of transient nonlinear reaction-convection-diffusion equations. Method of manufactured solution is used to test for correctness and bug-free implementation of the formulation. In the error analysis, optimal convergence is achieved. Applications in channel flow, cavity flow and predator-prey model is used to highlight the need and effectiveness of the stabilization technique

Four Point High Order Compact Iterative Schemes For The Solution Of The Helmholtz Equation

Teknik-teknik yang lebih baik diperoleh daripada beza terhingga dalam grid piawai dan grid putaran telah dibangunkan sejak beberapa tahun kebelakangan ini dalam menyelesaikan sistem linear yang terhasil daripada pendiskretan persamaan pembezaan separa (PDEs). Selain itu, satu sistem dengan peringkat kejituan yang lebih tinggi boleh dihasilkan daripada pendiskretan skema beza terhingga dengan menggunakan satu skim padat dengan kejituan peringkat empat yang dihasilkan daripada beza memusat dengan kejituan peringkat kedua. Dengan menggunakan beza terhingga padat ini, satu skim titik putaran dengan kejituan peringkat empat bagi persamaan Helmholtz dua dimensi (2D) yang baru terbentuk. Skim peringkat empat dalam grid piawai dan grid putaran boleh dikembangkan menjadi skim kumpulan ataupun sistem yang berperingkat empat. Sehubungan itu, kaedah multigrid berskala-multi digabungkan dengan ekstrapolasi Richardson diperkenalkan oleh Zhang [18] untuk menyelesaikan persamaan Poisson 2D. Improved techniques derived from the standard and rotated finite difference operators have been developed over the last few years in solving linear systems that arise from the discretization of various partial differential equations (PDEs) [14]. Furthermore, a higher order system can be generated from discretization of the finite difference scheme by using the fourth order compact scheme generated from the second order central difference. By using compact finite differences, new standard and rotated point schemes with fourth order accuracy for the two-dimensional (2D) Helmholtz equation are formulated in this thesis. The fourth order point schemes in both standard and rotated grids can be further applied to formulate a fourth order system to be used as group iterative method in their respective grid. On the other hand, the multiscale multigrid method combined with Richardson’s extrapolation is first introduced by Zhang [18] to solve the 2D Poisson equation

Numerical simulation using vorticity-vector potential formulation

An accurate and efficient computational method is needed for three-dimensional incompressible viscous flows in engineering applications. On solving the turbulent shear flows directly or using the subgrid scale model, it is indispensable to resolve the small scale fluid motions as well as the large scale motions. From this point of view, the pseudo-spectral method is used so far as the computational method. However, the finite difference or the finite element methods are widely applied for computing the flow with practical importance since these methods are easily applied to the flows with complex geometric configurations. However, there exist several problems in applying the finite difference method to direct and large eddy simulations. Accuracy is one of most important problems. This point was already addressed by the present author on the direct simulations on the instability of the plane Poiseuille flow and also on the transition to turbulence. In order to obtain high efficiency, the multi-grid Poisson solver is combined with the higher-order, accurate finite difference method. The formulation method is also one of the most important problems in applying the finite difference method to the incompressible turbulent flows. The three-dimensional Navier-Stokes equations have been solved so far in the primitive variables formulation. One of the major difficulties of this method is the rigorous satisfaction of the equation of continuity. In general, the staggered grid is used for the satisfaction of the solenoidal condition for the velocity field at the wall boundary. However, the velocity field satisfies the equation of continuity automatically in the vorticity-vector potential formulation. From this point of view, the vorticity-vector potential method was extended to the generalized coordinate system. In the present article, we adopt the vorticity-vector potential formulation, the generalized coordinate system, and the 4th-order accurate difference method as the computational method. We present the computational method and apply the present method to computations of flows in a square cavity at large Reynolds number in order to investigate its effectiveness

Invariant Discretization Schemes Using Evolution-Projection Techniques

Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy

A parallel compact-TVD method for compressible fluid dynamics employing shared and distributed-memory paradigms

A novel multi-block compact-TVD finite difference method for the simulation of compressible flows is presented. The method combines distributed and shared-memory paradigms to take advantage of the configuration of modern supercomputers that host many cores per shared-memory node. In our approach a domain decomposition technique is applied to a compact scheme using explicit flux formulas at block interfaces. This method offers great improvement in performance over earlier parallel compact methods that rely on the parallel solution of a linear system. A test case is presented to assess the accuracy and parallel performance of the new method