Efficient splitting schemes for magneto-hydrodynamic equations

We present in this paper several efficient numerical schemes for the magneto-hydrodynamic (MHD) equations. These semi-discretized (in time) schemes are based on the standard and rotational pressure-correction schemes for the Navier-Stokes equations and do not involve a projection step for the magnetic field. We show that these schemes are unconditionally energy stable, present an effective algorithm for their fully discrete versions and carry out demonstrative numerical experiments.     

Hybrid simulation of space plasmas: Models with massless fluid representation of electrons. II. Slow and intermediate shocks

This is a second article in a series of reviews on hybrid simulation of low-frequency processes in space plasmas. A hybrid model is described with ions represented by particles and electrons by a massless fluid. The main numerical schemes for the implementation of this model are described: the generalized Ohm law scheme and the predictor-corrector scheme. The first part of the article provides basic back-ground information: MHD models (ideal, resistive, and Hall model); the Rankine-Hugoniot relationship for MHD discontinuities; the Hoffman-Teller coordinate system; and a classification of discontinuities. The review part of the article surveys the literature on simulation of slow shocks (including switch-off shocks) and intermediate shocks. The survey of literature on hybrid simulation of intermediate shocks is concluded with a review of studies that use two different numerical codes (the hybrid model and the resistive Hall MHD model). The computation results produced by the two codes are compared. The concluding part presents some remarks concerning the existence of intermediate shocks and their relationship with rotational discontinuities in various numerical models (ideal MHD, resistive MHD, the hybrid model). Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 5–33, 1999.     

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L ² and H ¹ fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.     

Numerical solutions to the self‐similar transonic two‐dimensional nonlinear wave system

We present numerical results on a two‐dimensional Riemann problem governed by the self‐similar nonlinear wave system that gives rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain several numerical solutions for the model system. With the numerical results that we have obtained, we present several analyses of the transonic shock strengths and the positions of the transonic shocks with various Riemann data. Moreover, due to the presence of the corner of the object, numerical oscillations are apparent. We discuss regularity results for the solution near the corner of the object. Copyright © 2010 John Wiley & Sons, Ltd.     

Convergence of an unconditionally stable semi-implicit scheme for linear incompressible MHD equations

Semi-implicit methods have been introduced by Harned et al. to solve magneto-hydrodynamic equations (MHD) with numerical schemes which are unconditionally stable with respect to fast and shear Alfven modes. They prove the stability of their scheme for linear ideal MHD equations with periodic boundary conditions, and with some technical assumptions. In this paper, we prove convergence of the numerical approximation (time discretization), under the same hypothesis, but looking for solutions on any regular bounded open set of R³ with appropriate boundary conditions, and introducing finite resistivity and viscosity.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

A Well-Balanced HLL Scheme for Hyperbolic Conservation Systems With Source Terms北大核心CSCD

针对含源项的双曲守恒方程给出了一种新的有限体积格式.经典的有限体积格式不能正确地模拟对流通量项和外力之间的平衡所产生的动力学问题.为解决这个问题,仿照经典的HLL近似Riemann求解器设计思路设计了含源项的近似Riemann求解器.针对含重力源项的一维流体Euler方程和理想磁流体方程,通过对通量计算格式的修正得到了保平衡HLL格式（WB-HLL）,并给出了保平衡的证明.针对一维Euler方程和理想磁流体给出了两个算例,比较了传统HLL格式和提出的WB-HLL格式的计算精度.计算结果表明,WB-HLL格式精度更高,收敛更快.     

Investigation of semi-implicit schemes for the solution of stiff systems of MHD equations

An efficient semi-implicit method for integration of stiff systems of MHD equations is considered. A comparison is made of the numerical results obtained for the solution of the full linear system of MHD equations in cylindrical geometry using explicit and semi-implicit schemes.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 208–214, 1993.     

Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

In this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.     

Structure-preserving space-time discretization of nonlinear structural dynamics based on a mixed variational formulation

The present work deals with the design of structure-preserving numerical methods in the field of nonlinear elastodynamics and structural dynamics. Structure-preserving schemes such as energy-momentum consistent (EMC) methods are known to exhibit superior numerical stability and robustness. Most of the previously developed schemes are relying on a displacement-based variational formulation of the underlying mechanical model. In contrast to that we present a mixed variational framework for the systematic design of EMC schemes. The newly proposed mixed approach accomodates high-performance mixed finite elements such as the shell element due to Wagner & Gruttmann [1] and the brick element due to Kasper & Taylor [2]. Accordingly, the proposed approach makes possible the structure-preserving extension to the dynamic regime of those high-performance elements. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On stability of Alfvén discontinuities

We study Alfvén discontinuities for the equations of ideal compressible magnetohydrodynamics (MHD). The Alfvén discontinuity is a characteristic discontinuity for the hyperbolic system of the MHD equations but, as for shock waves, the gas crosses its front. By numerical testing of the Lopatinskii condition, we carry out spectral stability analysis, i.e. we find the parameter domains of stability and violent instability of planar Alfvén discontinuities. We also show that Alfvén discontinuities can be only weakly stable in the sense that the uniform Lopatinskii condition is never satisfied. Copyright © 2008 John Wiley & Sons, Ltd.     

具弱色散误差的单调差分格式

本文基于[1]、[2]的理论分析,分别给出了计算(一维、二维)非线性双曲方程组的两类具弱色散误差的单调格式。§2针对一维气动力学方程组,通过数值结果。就激波的逼近问题,与已有的格式进行了比较。     

Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see, e.g., [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.     

On the use of Mühlbach expansions in the recovery step of ENO methods

Summary. The recovery step is the most expensive algorithmic ingredient in modern essentially non-oscillatory (ENO) shock capturing methods on triangular meshes for the numerical simulation of compressible fluid flow. While recovery polynomials in Newton form are used in one-dimensional ENO schemes it is a priori not clear whether such useful as well as numerically stable form of polynomials exists in multiple dimensions. As was observed in [1] a very general answer to this question was provided by Mühlbach in two subsequent papers [15] and [16]. We generalise his interpolation theory further to the general recovery problem and outline the use of Mühlbach's expansion in ENO schemes. Numerical examples show the usefulness of this approach in the problem of recovery from cell average data. Received August 24, 1995 / Revised version received December 14, 1995     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

On the consistent discretization in time of nonlinear thermo-elastodynamics

The main goal of the present work is to describe two structure-preserving time integrators, which have been proposed by Romero [1] (Thermodynamically Consistent (TC) algorithm) and Groß [2] (enhanced hybrid Galerkin (ehG) method) and to compare them with standard integrators. The numerical results of the two schemes are compared in the context of a thermoelastic two-dimensional double pendulum. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improvement of convergence to steady state solutions of Euler equations with weighted compact nonlinear schemes

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22-44 and Zhang S., Jiang S. and Shu C.-W. (2008), J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273-305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216-238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.     

Nonlinear generalized functions and jump conditions for a standard one pressure liquid–gas model

The mixture of a liquid and a gas is classically represented by one pressure models. These models are a system of PDEs in nonconservative form and shock wave solutions do not make sense within the theory of distributions: they give rise to products of distributions that are not defined within distribution theory. But they make sense by applying a theory of nonlinear generalized functions to these equations. In contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori. Jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations. The physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible, and so they do not satisfy equally well their respective state laws on the shock waves. This modelization gives well defined explicit jump conditions. The great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results. The jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions, which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results. We expose with care the mathematical originality of the theory of nonlinear generalized functions (an original abstract analysis issued by the Leopoldo Nachbin team on infinite dimensional holomorphy) that permits to state mathematically physical facts that cannot be formulated within distribution theory, and are the key for the removal of “ambiguities” that classically appear when one tries to calculate on “multiplications of distributions” that occur in the differential equations of physics.     

Hybrid WENO schemes with different indicators on curvilinear grids

In {J. Comput. Phys. 229 (2010) 8105-8129}, we studied hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for hyperbolic conservation laws on uniform grids for Cartesian domains. In this paper, we extend the schemes to solve two-dimensional systems of hyperbolic conservation laws on curvilinear grids for non-Cartesian domains. Our goal is to obtain similar advantageous properties as those of the hybrid WENO schemes on uniform grids for Cartesian domains. Extensive numerical results strongly support that the hybrid WENO schemes with discontinuity indicators on curvilinear grids can also save considerably on computational cost in contrast to the pure WENO schemes. They also maintain the essentially non-oscillatory property for general solutions with discontinuities and keep the sharp shock transition.