On the Solution of Convection-Diffusion Boundary Value Problems Using Equidistributed Grids

The effect of using grid adaptation on the numerical solution of model convection-diffusion equations with a conservation form is studied. The grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc-length of the solution. In particular, a parameter-dependent monitor function is introduced which incorporates fixed meshes, approximate arc-length equidistribution, and equidistribution of the absolute value of the solution, in a single framework. Thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values. A class of singularly perturbed problems, including Burgers's equation in the limit of small viscosity, is studied. Singular perturbation and bifurcation techniques are used to analyze the solution of the discretized equations, and numerical results are compared with the results from the analysis. Computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory. It is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that, furthermore, the spurious solutions can be stable for an appropriate moving mesh method

Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters

Preprint version, the final publication is available at Springer via http://dx.doi.org/10.1007/s10543-015-0559-8This paper discusses the numerical solution of 1-D convection-diffusion-reaction problems that are singularly perturbed with two small parameters using a new mesh-adaptive upwind scheme that adapts to the boundary layers. The meshes are generated by the equidistribution of a special positive monitor function. Uniform, parameter independent convergence is shown and holds even in the limit that the small parameters are zero. Numerical experiments are presented that illustrate the theoretical findings, and show that the new approach has better accuracy compared with current methods.DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamic

Parameter-uniform numerical method for global solution and global normalized flux of singularly perturbed boundary value problems using grid equidistribution

AbstractIn this paper, we present the analysis of an upwind scheme for obtaining the global solution and the normalized flux for a convection–diffusion two-point boundary value problem. The solution of the upwind scheme is obtained on a suitable nonuniform mesh which is formed by equidistributing the arc-length monitor function. It is shown that the discrete solution obtained by the upwind scheme and the global solution obtained via interpolation converges uniformly with respect to the perturbation parameter. In addition, we prove the uniform first-order convergence of the weighted derivative of the numerical solution on this nonuniform mesh and the uniform convergence of the global normalized flux on the whole domain. Numerical results are presented that demonstrate the sharpness of our results

A simple adaptive grid method in two dimensions

This is the published version, also available here: http://dx.doi.org/10.1137/0915049.This paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems. It is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and, based on this recognition, a local equidistribution principle is developed. A discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality. The adaptive grid method that is constructed contains three grid-quality parameters. Numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations. Results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers' equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers' equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution

A volume-preserving sharpening approach for the propagation of sharp phase boundaries in multiphase lattice Boltzmann simulations

Lattice Boltzmann models that recover a macroscopic description of multiphase flow of immiscible liquids typically represent the boundaries between phases using a scalar function, the phase field, that varies smoothly over several grid points. Attempts to tune the model parameters to minimise the thicknesses of these interfaces typically lead to the interfaces becoming fixed to the underlying grid instead of advecting with the fluid velocity. This phenomenon, known as lattice pinning, is strikingly similar to that associated with the numerical simulation of conservation laws coupled to stiff algebraic source terms. We present a lattice Boltzmann formulation of the model problem proposed by LeVeque and Yee [J. Comput. Phys. 86, 187] to study the latter phenomenon in the context of computational combustion, and offer a volume-conserving extension in multiple space dimensions. Inspired by the random projection method of Bao and Jin [J. Comput. Phys. 163, 216] we further generalise this formulation by introducing a uniformly distributed quasi-random variable into the term responsible for the sharpening of phase boundaries. This method is mass conserving and the statistical average of this method is shown to significantly delay the onset of pinning