Mjh Martijn Anthonissen

The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.

Extension of the Complete Flux Scheme to Time-Dependent Conservation Laws

We present the stationary and transient complete flux schemes for the advection-diffusion-reaction equation. In the first scheme, the numerical flux is derived from a local BVP for the stationary equation. The transient scheme is an extension, since it includes the time derivative in the flux computation. The resulting semidiscretization is an implicit ODE system, which has much smaller dissipation and dispersion errors than the semidiscretization based on the stationary flux, at least for smooth problems. Both schemes are validated for a test problem.

Local Defect Correction Techniques Applied to a Combustion Problem

The standard local defect correction (LDC) method has been extended to include multilevel adaptive gridding, domain decomposition, and regridding. The domain decomposition algorithm provides a natural route for parallelization by employing many small tensor-product grids, rather than a single large unstructured grid. The algorithm is applied to a laminar Bunsen flame with one-step chemistry.

The Complete Flux Scheme for Spherically Symmetric Conservation Laws

We apply the finite volume method to a spherically symmetric conservation law of advection-diffusion-reaction type. For the numerical flux we use the so-called complete flux scheme. In this scheme the flux is computed from a local boundary value problem for the complete equation, including the source term. As a result, the numerical flux is the superposition of a homogeneous flux and an inhomogeneous flux. The resulting scheme is second order accurate, uniformly in the Peclet numbers.

Numerical Dissipation and Dispersion of the Homogeneous and Complete Flux Schemes

We analyse numerical dissipation and dispersion of the homogeneous flux (HF) and complete flux (CF) schemes, finite volume methods introduced in [4]. To that purpose we derive the modified equation of both schemes . We show that the HF scheme suffers from numerical diffusion for dominant advection, which is effectively removed in the CF scheme. The latter scheme, however, is prone to numerical dispersion. We validate both schemes for a model problem.

The Complete Flux Scheme for Conservation Laws in Curvilinear Coordinates

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Real-time control of surface remelting

We consider a model for laser surface remelting, a process to improve the surface quality of steel components. The mathematical model consists of the two-dimensional heat equation for temperature and an ordinary differential equation for the liquid phase. The equations are coupled via source terms. We study the efficient numerical simulation using adaptive grids, which are especially well-suited for problems with moving heat sources. To account for the local high activity due to the heat source, we introduce local uniform grids and couple the solutions on the global coarse and local fine grids using the local defect correction (LDC) technique.

Mimetic staggered discretization of incompressible navier–Stokes for barycentric dual mesh

A staggered discretization of the incompressible Navier–Stokes equations is presented for polyhedral non orthogonal nonsmooth meshes admitting a barycentric dual mesh. The discretization is constructed by using concepts of discrete exterior calculus. The method strictly conserves mass, momentum and energy in the absence of viscosity.

Discretization and parallel iterative schemes for advection-diffusion-reaction problems

Conservation laws of advection-diffusion-reaction (ADR) type are ubiquitous in continuum physics. In this paper we outline discretization of these problems and iterative schemes for the resulting linear system. For discretization we use the finite volume method in combination with the complete flux scheme. The numerical flux is the superposition of a homogeneous flux, corresponding to the advection-diffusion operator, and the inhomogeneous flux, taking into account the effect of the source term (ten Thije Boonkkamp and Anthonissen, J Sci Comput 46(1):47–70, 2011). For a three-dimensional conservation law this results in a 27-point coupling for the unknown as well as the source term. Direct solution of the sparse linear systems that arise in 3D ADR problems is not feasible due to fill-in. Iterative solution of such linear systems remains to be the only efficient alternative which requires less memory and shorter time to solution compared to direct solvers. Iterative solvers require a preconditioner to reduce the number of iterations. We present results using several different preconditioning techniques and study their effectiveness.

A monolithic fluid-structure interaction method, application to a piston problem

A monolithic FSI method is presented. A standard piston problem is considered as test case. The piston problems fluid domain is represented by a closed tube filled with air. One end of the fluid tube is formed by a piston connected to a spring. We use the Euler equations of gas dynamics as well as a linear simplification of these, the acoustic equations, to model the gas dynamics in the tube. A Discontinuous Galerkin method is applied to discretize the fluidflow equations, together with an immersed-boundary method to account for the moving piston. A monolithic formulation of the coupled system is derived and analyzed. It is proven that the semidiscrete formulation is stable, if two correction terms are used at the coupling interface. We use Lyapunov functions to prove stability of the semi-discrete monolithic formulation. Further, different time-integration methods are considered, analyzed and tested. The numerical results are very accurate; they correspond very well to analytical approximations. The theoretical prediction of the eigenfrequency can be reproduced very accurately. Moreover, the amplitude of the spring oscillation is conserved very well.