M.J.H. Anthonissen

Convergence analysis of the local defect correction method for diffusion equations

Summary.This paper is concerned with the convergence analysis of the local defect correction (LDC) method for diffusion equations. We derive a general expression for the iteration matrix of the method. We consider the model problem of Poissons equation on the unit square and use standard five-point finite difference discretizations on uniform grids. It is shown via both an upper bound for the norm of the iteration matrix and numerical experiments, that the rate of convergence of the LDC method is proportional to H2 with H the grid size of the global coarse grid.

Local defect correction for time-dependent partial differential equations

A Local Defect Correction (LDC) method for solving time-dependent partial differential equations whose solutions have highly localized properties is discussed. We present some properties of the technique. Results of numerical experiments illustrate the accuracy and the efficiency of the method.

A Two-Dimensional Complete Flux Scheme in Local Flow Adapted Coordinates

We present a formulation of the two-dimensional complete flux (CF) scheme in terms of local orthogonal coordinates adapted to the flow, i.e., one coordinate axis is aligned with the local velocity field and the other one is perpendicular to it. This approach gives rise to an advection-diffusion-reaction boundary value problem (BVP) for the flux component in the local flow direction. For the other (diffusive) flux component we use central differences. We will demonstrate the performance of the scheme for several examples.

A Compact High Order Finite Volume Scheme for Advection‐Diffusion‐Reaction Equations

We present a new integral representation for the flux of the advection‐diffusion‐reaction equation, which is based on 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 Gauss‐Legendre quadrature rules to the integral representation gives the high order finite volume complete flux scheme, which is fourth order accurate for both diffusion dominated and advection dominated flow.

Modeling liquid slugs accelerating in inclined conduits

In this article, we simulate traveling liquid slugs in conduits, as they may occur in systems carrying high-pressure steam. We consider both horizontal and inclined pipes in which the slug is accelerated by a suddenly applied pressure gradient, while at the same time, gravity and friction work in the opposite direction. This causes a steep slug front and an extended slug tail. The shapes of front and tail are of interest since they determine the forces exerted on bends and other obstacles in the piping system. The study also aims at improving existing one-dimensional (1D) models. A hybrid model is proposed that enables us to leave out the larger inner part of the slug. It was found that the hybrid model speeds up the two-dimensional (2D) computations significantly, while having no adverse effects on the shapes of the slugs front and tail.

Compact high order complete flux schemes

In this paper we outline the complete flux scheme for an advection-diffusion-reaction model problem. The scheme is based on the integral representation of the flux, which we derive from a local boundary value problem for the entire equation, including the source term. Consequently, the flux consists of a homogeneous part, corresponding to the advection-diffusion operator, and an inhomogeneous part, taking into account the effect of the source term. We apply (weighted) Gauss quadrature rules to derive the standard complete flux scheme, as well as a compact high order variant. We demonstrate the performance of both schemes.

LDC with compact FD schemes for convection-diffusion equations

We discuss an algorithm for convection-diffusion equations with high activity areas which combines the Local Defect Correction technique with high order compact finite difference schemes.