A Roe Scheme for Ideal MHD Equations on 2D Adaptively Refined Triangular Grids

In this paper we present a second order finite volume method for the resolution of the bidimensional ideal MHD equations on adaptively refined triangular meshes. Our numerical flux function is based on a multidimensional extension of the Roe scheme proposed by Cargo and Gallice for the 1D MHD system. If the mesh is only composed of triangles, our scheme is proved to be weakly consistent with the condition …B=0. This property fails on a cartesian grid. The efficiency of our refinement procedure is shown on 2D MHD shock capturing simulations. Numerical results are compared in case of the interaction of a supersonic plasma with a cylinder on the adapted grid and several non-refined grids. We also present a mass loading simulation which corresponds to a 2D version of the interaction between the solar wind and a comet.     

High-Order Scheme for a Nonlinear Maxwell System Modelling Kerr Effect

This paper is devoted to the derivation of an efficient numerical scheme for the Kerr–Maxwell system. We begin by studying the 1-D Riemann problem. We obtain a result of existence and uniqueness for large data. Then we develop a high-order Roe solver and exhibit solutions in 1-D and 2-D simulations.     

A Residual-Based Compact Scheme for the Compressible Navier–Stokes Equations

A simple and efficient time-dependent method is presented for solving the steady compressible Euler and Navier–Stokes equations with third-order accuracy. Owing to its residual-based structure, the numerical scheme is compact without requiring any linear algebra, and it uses a simple numerical dissipation built on the residual. The method contains no tuning parameter. Accuracy and efficiency are demonstrated for 2-D inviscid and viscous model problems. Navier–Stokes calculations are presented for a shock/boundary layer interaction, a separated laminar flow, and a transonic turbulent flow over an airfoil.     

High-Order Nonreflecting Boundary Conditions without High-Order Derivatives

A wave problem in an unbounded domain is often treated numerically by truncating the infinite domain via an artificial boundary , imposing a so-called nonreflecting boundary condition (NRBC) on , and then solving the problem numerically in the finite domain bounded by . A general approach is devised here to construct high-order local NRBCs with a symmetric structure and with only low (first- or second-) order spatial and/or temporal derivatives. This enables the practical use of NRBCs of arbitrarily high order. In the case of time-harmonic waves with finite element discretization, the approach yields a symmetric C⁰ finite element formulation in which standard elements can be employed. The general methodology is presented for both the time-harmonic case (Helmholtz equation) and the time-dependent case (the wave equation) and is demonstrated numerically in the former case.     

A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.     

A Triangulated Vortex Method for the Axisymmetric Euler Equations

A new method is presented for the prediction of unsteady axisymmetric inviscid flows. By combining a triangulated vortex approach with a novel evaluation technique for the Biot–Savart integrals, a Lagrangian vortex method is developed which eliminates the singularities usually present in axisymmetric methods, without recourse to normalizations or other approximations. Furthermore, the computational effort scales as the number of control points N and, in the large N limit, depends only on the order of quadrature employed. The accuracy and computational effort are assessed by comparison with the velocity field of a Gaussian core vortex ring and the use of the technique is illustrated by computation of the motion of Norbury rings and of vortex ring pairing.     

New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

We introduce a new high-resolution central scheme for multidimensional Hamilton–Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/Δt. By letting Δt↓0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge–Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.     

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.     

A General-Purpose Finite-Volume Advection Scheme for Continuous and Discontinuous Fields on Unstructured Grids

We present a new general-purpose advection scheme for unstructured meshes based on the use of a variation of the interface-tracking flux formulation recently put forward by O. Ubbink and R. I. Issa (J. Comput. Phys.153, 26 (1999)), in combination with an extended version of the flux-limited advection scheme of J. Thuburn (J. Comput. Phys.123, 74 (1996)), for continuous fields. Thus, along with a high-order mode for continuous fields, the new scheme presented here includes optional integrated interface-tracking modes for discontinuous fields. In all modes, the method is conservative, monotonic, and compatible. It is also highly shape preserving. The scheme works on unstructured meshes composed of any kind of connectivity element, including triangular and quadrilateral elements in two dimensions and tetrahedral and hexahedral elements in three dimensions. The scheme is finite-volume based and is applicable to control-volume finite-element and edge-based node-centered computations. An explicit–implicit extension to the continuous-field scheme is provided only to allow for computations in which the local Courant number exceeds unity. The transition from the explicit mode to the implicit mode is performed locally and in a continuous fashion, providing a smooth hybrid explicit–implicit calculation. Results for a variety of test problems utilizing the continuous and discontinuous advection schemes are presented.     

A Particle-Mesh Method for the Shallow Water Equations Near Geostrophic Balance

In this paper we outline a new particle-mesh method for rapidly rotating shallow water flows based on a set of regularized equations of motion. The time-stepping method uses an operator splitting of the equations into an Eulerian gravity wave part and a Lagrangian advection part. An essential ingredient is the advection of absolute vorticity by means of translated radial basis functions. We show that this implies exact conservation of enstrophy. The method is tested on two model problems based on the qualitative features of the solutions obtained (i.e., dispersion or smoothness of potential vorticity contours) as well as on the increase in mean divergence level.     

Half-Moment Closure for Radiative Transfer Equations

In this paper a moment method for radiative transfer equations is considered which has been developed and investigated using different approaches. Problems appearing for this moment system for boundary value problems using Maxwell-type boundary conditions are described. A new method based on the consideration of positive and negative half fluxes is developed and shown to overcome the above problems. Moreover, a numerical scheme and numerical results for the new moment system are presented.     

A Composite Scheme for Gas Dynamics in Lagrangian Coordinates

One cycle of a composite finite difference scheme is defined as several time steps of an oscillatory scheme such as Lax–Wendroff followed by one step of a diffusive scheme such as Lax–Friedrichs. We apply this idea to gas dynamics in Lagrangian coordinates. We show numerical results in two dimensions for Noh's infinite strength shock problem and the Sedov blast wave problem, and for several one-dimensional problems including a Riemann problem with a contact discontinuity. For Noh's problem the composite scheme produces a better result than that obtained with a more conventional Lagrangian code.     

Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattices

In this paper, a lattice Boltzmann (LB) scheme for convection diffusion on irregular lattices is presented, which is free of any interpolation or coarse graining step. The scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the Maxwell–Boltzmann distribution. The axioma holds for both Bravais and irregular lattices, implying a single framework for LB schemes for all lattice types. By solving benchmark problems we have shown that the scheme is indeed consistent with convection diffusion. Furthermore, we have compared the performance of the LB schemes with that of finite difference and finite element schemes. The comparison shows that the LB scheme has a similar performance as the one-step second-order Lax–Wendroff scheme: it has little numerical diffusion, but has a slight dispersion error. By changing the relaxation parameter ω the dispersion error can be balanced by a small increase of the numerical diffusion.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.     

Unconditionally Stable Methods for Hamilton–Jacobi Equations

We present new numerical methods for constructing approximate solutions to the Cauchy problem for Hamilton–Jacobi equations of the form u_t+H(D_xu)=0. The methods are based on dimensional splitting and front tracking for solving the associated (non-strictly hyperbolic) system of conservation laws p_t+D_xH(p)=0, where p=D_xu. In particular, our methods depend heavily on a front tracking method for one-dimensional scalar conservation laws with discontinuous coefficients. The proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as “large-time-step” Godunov-type (or front tracking) methods. We present several numerical examples illustrating the main features of the proposed methods. We also compare our methods with several methods from the literature.     

Comparison of Several Spatial Discretizations for the Navier–Stokes Equations

Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier–Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourth-order differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.     

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.     

On Backtracking Failure in Newton–GMRES Methods with a Demonstration for the Navier–Stokes Equations

In an earlier study of inexact Newton methods, we pointed out that certain counterintuitive behavior may occur when applying residual backtracking to the Navier–Stokes equations with heat and mass transport. Specifically, it was observed that a Newton–GMRES method globalized by backtracking (linesearch, damping) may be less robust when high accuracy is required of each linear solve in the Newton sequence than when less accuracy is required. In this brief discussion, we offer a possible explanation for this phenomenon, together with an illustrative numerical experiment involving the Navier–Stokes equations.     

Introducing Physical Relaxation Terms in Bloch Equations

The Bloch equation models the evolution of the state of electrons in matter described by a Hamiltonian. To model more physical phenomena we have to introduce phenomenological relaxation terms. The introduction of these terms has to conserve some positiveness properties. The aim of this paper is to review possible relaxation models and to provide insight into how to discretize them properly in view of numerical computations.     

The Three Dimensional Non-conforming Finite Element Solution of the Chapman–Ferraro Problem

We demonstrate the feasibility of using a non-conforming, piecewise harmonic finite element method on an unstructured grid in solving a magnetospheric physics problem. We use this approach to construct a global discrete model of the magnetic field of the magnetosphere that includes the effects of shielding currents at the outer boundary (the magnetopause). As in the approach of F. R. Toffolettoet al.(1994,Geophys. Res. Lett.21, 7) the internal magnetospheric field model is that of R. V. Hilmer and G.-H. Voigt (1995,J. Geophys. Res.) while the magnetopause shape is based on an empirically determined approximation (1997, J. Shueet al.,J. Geophys. Res.102, 9497). The results is a magnetic field model whose field lines are completely confined within the magnetosphere. The presented numerical results indicate that the discrete non-conforming finite element model is well-suited for magnetospheric field modeling.