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

Calculations in Mathematica on low-frequency diffraction by a circular disk

This paper is devoted to the symbolic calculation of the scattering coefficient in diffraction by a circular disk, by the use of Mathematica. Three diffraction problems are considered: scalar diffraction by an acoustically soft disk, scalar diffraction by an acoustically hard disk, and electromagnetic diffraction by a perfectly conducting disk. In the low-frequency approximation, the solutions of these problems are in the form of expansions in powers of ka, where a is the radius of the disk and k is the wave number. The emphasis is on the low-frequency expansion for the scattering coefficient, of which several terms are determined exactly with the help of Mathematica

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

Convergence properties of local defect correction algorithm for the boundary element method

\u3cp\u3eSometimes boundary value problems have isolated regions where the solution changes rapidly. Therefore, when solving numerically, one needs a fine grid to capture the high activity. The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid. One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique. The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid. The algorithm is relatively new and its convergence properties have not been studied for the boundary element method. In this paper the objective is to determine convergence properties of the algorithm for the boundary element method. First, we formulate the algorithm as a fixed point iterative scheme, which has also not been done before for the boundary element method, and then study the properties of the iteration matrix. Results show that we can always expect convergence. Therefore, the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.\u3c/p\u3

Compact high order complete flux schemes

\u3cp\u3eIn 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.\u3c/p\u3

An improved corrective smoothed particle method approximation for second‐order derivatives

To solve (partial) differential equations it is necessary to have good numerical approximations. In SPH, most approximations suffer from the presence of boundaries. In this work a new approximation for the second-order derivative is derived and numerically compared with two other approximation methods for a simple test case. The new method is slightly more expensive, but leads to a significantly improved accuracy

A locally refined cut-cell method with exact conservation for the incompressible Navier-Stokes equations

\u3cp\u3eWe present a mimetic discretization of the incompressible Navier-Stokes equations for general polygonal meshes. The discretization employs staggered velocity variables and results in discrete equations that exactly conserve mass, momentum and kinetic energy (in the inviscid limit) up to and including the boundaries where Dirichlet conditions apply. Moreover, the discrete equations give rise to a discrete global vorticity that is consistent with the Dirichlet boundary conditions. As the method retains all its favorable properties on general meshes, it can be perfectly applied as a locally refined Cartesian mesh cut-cell method. We numerically verify the conservation properties for the lid-driven cavity flow and demonstrate the method for the unsteady flow around a circular cylinder.\u3c/p\u3

An improved corrective smoothed particle method approximation for second-order derivatives

To solve (partial) differential equations it is necessary to have good numerical approximations. In SPH, most approximations suffer from the presence of boundaries. In this work a new approximation for the second-order derivative is derived and numerically compared with two other approximation methods for a simple test case. The new method is slightly more expensive, but leads to a significantly improved accuracy

Conservative polytopal mimetic discretization of the incompressible Navier–Stokes equations

\u3cp\u3eWe discretize the incompressible Navier–Stokes equations on a polytopal mesh by using mimetic reconstruction operators. The resulting method conserves discrete mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. To do this we introduce a dual mesh and show how the dual mesh can be completed to a cell-complex. We present existing mimetic reconstruction operators in a new symmetric way applicable to arbitrary dimension, use these to interpolate between primal and dual mesh and derive properties of these operators. Finally, we test both 2- and 3-dimensional versions of the method on a variety of complicated meshes to show its wide applicability. We numerically test the convergence of the method and verify the derived conservation statements.\u3c/p\u3