Practical engineering problems in heat transfer and fluid flow involve one or more governing equations, together with some boundary conditions over a domain. The domain is often complex and non-uniform. The ability to use a mesh of finite elements to accurately discretise domain of any size and shape makes the finite element method a powerful tool to numerically analyse problems in these areas....

We establish a general framework for analyzing the class of finite volume methods which employ continuous or totally discontinuous trial functions and piecewise constant test functions. Under the framework, optimal order convergence in the H1 and L2 norms can be obtained in a natural and systematic way for classical finite volume methods and new finite volume methods such as discontinuous finit...

Abstract. A semidiscrete finite volume element (FVE) approximation to a parabolic integrodifferential equation (PIDE) is analyzed in a two-dimensional convex polygonal domain. An optimalorder L2-error estimate for smooth initial data and nearly the same optimal-order L2-error estimate for nonsmooth initial data are obtained. More precisely, for homogeneous equations, an elementary energy techni...

Abstract. In this paper, we present a new type of Hermite weighted essentially nonoscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations on the finite volume framework. The cell averages of the function and its first one (in one dimension) or two (in two dimensions) derivative values are together evolved via time approaching and used in the reconstructions. And the major advantag...

We examine a cell-vertex finite volume method which is applied to a model parabolic convection-diffusion problem. By using techniques from finite element analysis, local errors away from all layers are obtained in a seminorm that is related to, but weaker than, the L2 norm.

Vorticity-stream function method and MAC algorithm are adopted to systemically compare the finite volume method (FVM) and finite difference method (FDM) in this paper. Two typical problems—lid-driven flow and natural convection flow in a square cavity—are taken as examples to compare and analyze the calculation performances of FVM and FDMwith variant mesh densities, discrete forms, and treatmen...

We study a very simple model for nucleation-and-growth in infinite volume in the low-temperature limit. Despite its simplicity, this model exhibits a very rich metastable behavior. Depending on the speed of growth, the system goes trough four different regimes: 1) both the “shape of the critical droplet” and the typical relaxation time are the same as in finite volume. 2) the “shape of the crit...

Mathematical models of transport of radioactive contaminants in flowing groundwater involve large systems of coupled advection dominated transport equations. High-resolution explicit finite volume methods, if applied to advective part of model and combined with appropriate numerical methods for diffusion-dispersionreaction part, can offer precise and monotone numerical solutions, but they requi...