We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation mod-eling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semi-implicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time step size. Owing to energy stability, we show that the scheme is stable in the time and space discrete and norms. We also present an efficient, practical nonlinear multigrid method . comprised of a standard FAS method for the Cahn-Hilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part . for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy- and boundary-driven flows.

We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

In this paper we propose a development of the finite difference method, called the tailored finite point method, for solving steady magnetohydrodynamic (MHD) duct flow problems with a high Hartmann number. When the Hartmann number is large, the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer. Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy. However, the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh. Firstly, we devise the tailored finite point method for the scalar inhomogeneous convection-diffusion problem, and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations. For each interior grid point of a given rectangular mesh, we construct a finite-point difference operator at that point with some nearby grid points, where the coefficients of the difference operator are tailored to some particular properties of the problem. Numerical examples are provided to show the high performance of the proposed method.

This paper describes the extension of arecently developed numerical solver for the Landau-LifshitzNavier-Stokes (LLNS) equations to binary mixtures in threedimensions. The LLNS equations incorporate thermal fluctuations intomacroscopic hydrodynamics by using white-noise fluxes. Thesestochastic PDEs are more complicated in three dimensions due to thetensorial form of the correlations for the stochastic fluxes and inmixtures due to couplings of energy and concentration fluxes (e.g.,Soret effect). We present various numerical tests of systems in andout of equilibrium, including time-dependent systems, anddemonstrate good agreement with theoretical results and molecularsimulation.

We discuss the occurrence of oscillationswhen using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered andnon-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema.The dependence of oscillatory properties on the numerical viscositycoefficient is investigated rigorously for the LFt schemes, illuminating alsothe properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient areprone to oscillations at data extrema. For all LFt schemes except for the classicalLax-Friedrichs method, occurring oscillations are damped in the course of a computation.This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggeredversion may yield oscillatory results, while it can be shown rigorously that the staggered NTscheme does not produce oscillations when using the classical minmod-limiter under arestriction on the time step size. Note that this restriction is not thesame as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper.

We are concerned with the structure of the operator corresponding to the Lax–Friedrichs method. At first, the phenomenae which may arise by the naive use of the Lax–Friedrichs scheme are analyzed.In particular, it turns out that the correct definition of the method has to include the details of the discretization of the initial condition and the computational domain. Based on the results of thediscussion, we give a recipe that ensures that the number of extrema within the discretized version of the initial data cannot increase by the application of the scheme. The usefulness of the recipe is confirmed by numerical tests.

Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference schemes in the stationary case with constant coefficients and in the time-dependent case with variable coefficients by using control theory and probabilistic methods. In this paper we are able to handle variable coefficients by a purely analytical method. In our opinion this way is far simpler and, for the cases we can treat, it yields a better rate of convergence than Krylov obtains in the variable coefficients case.

In order to accommodate solutions with multiple phases, corresponding to crossing rays, weformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from anassociated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based ondelta and Heaviside functions and analyze the resultingequations. They form systems of nonlinear conservation laws with source terms. In contrast to the classicaleikonal equation, theseequations will incorporate a "finite" superposition principle in the sense that while the maximum number of phasesis not exceeded a sum of solutions is also a solution. We present numerical results for a varietyof homogeneous and inhomogeneous problems.

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.