A fractional step ELLAM approach to high-dimensional convection–diffusion problems with forward particle tracking

Abstract

In this paper, a fractional step method combined with an Eulerian–Lagrangian localized adjoint method (ELLAM) is proposed to solve high-dimensional convection–diffusion problems. The method reduces high-dimensional problems to a series of uncoupled one-dimensional problems in each time step interval, in which one-dimensional ELLAM is used to solve the one-dimensional splitting equations. The approach takes the attractive advantages of the ELLAM method and the fractional step technique. It reduces computational complexities, large memory requirements, and long computation durations due to the application of the splitting technique. It reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation. It effectively eliminates non-physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way. Numerical experiments show the efficient performance of the approach.

Introduction

Unsteady convection–diffusion equations that involve a combination of advection and diffusion dynamical processes are among the most widespread in various areas of science and technology, e.g., heat and mass transfer, oil reservoir simulation, groundwater modelling, and aerodynamics and physiology (see, for example [1], [2], [11], [21]). In many such applications, the convection terms essentially dominate the diffusion terms, which leads to a nearly hyperbolic set of governing partial differential equations. The numerical approximation to the problems presents a challenging computational task. It is well documented that when the governing equation is convection-dominated, many standard numerical methods developed for diffusion-dominated processes often exhibit some combination of difficulties ranging from non-physical oscillations (central difference/Galerkin scheme) to excessive numerical diffusions (upstream scheme) at steep fronts.

Many works have been done to overcome these difficulties and to allow accurate numerical solutions with reasonable computational efforts. One class of approximations is the Eulerian method. In the framework of the Eulerian approach, the traditional finite difference and finite element methods are improved, e.g., the flux corrected transport scheme, total variation diminishing scheme, streamline-upwind Petrov–Galerkin method, and optimal test function method. However, the methods in the Eulerian framework are limited by Courant number restrictions. A second class of approximations is based on treatment of the hyperbolic part by the Lagrangian method, in which the remainder of the equation is treated by an Eulerian-type approximation. While these methods reduce temporal errors and overcome the Courant number restrictions, most of them do not conserve mass. To overcome the limitations associated with these Eulerian–Lagrangian approximations, Celia et al. [4] proposed the Eulerian–Lagrangian localized adjoint method (ELLAM) for one-dimensional convection–diffusion problems. The ELLAM formulation provides a general characteristic solution procedure and a consistent framework for conserving mass and treating boundary conditions. It overcomes the principal shortcoming of some characteristic methods while maintaining their numerical advantages. It reduces temporal errors and therefore allows for large time step sizes in computation without the loss of accuracy, and is highly resistant to numerical dispersion in the presence of small dispersivities. This method is an accurate and efficient solver of the linear convection-dominated diffusion problems with large Courant numbers. The further study of the ELLAM has been successfully taken for high-dimensional problems by Binning and Celia [3], Healy and Russell [13], and Wang et al. [26]. Furthermore, the ELLAM technique combined with the mixed finite element method has successfully been developed to solve the miscible fluid flows in porous media with point sources and sinks in [27], [28], which can accurately simulate incompressible and compressible fluid flows in porous media for oil reservoir simulation, as well as for the highly compressible multicomponent fluid flows in porous media.

Due to the complexities and huge computational costs in realistic long term and large scale simulations, there is strong interest in developing an efficient solution technique. This motivates us to study a fractional step ELLAM method (FS-ELLAM) to solve multi-dimensional convection–diffusion problems. The fractional step method is an efficient numerical technique for solving multi-dimensional parabolic problems (see, for example [5], [6], [8], [9], [10], [12], [14], [16], [17], [19], [23], [24], [25]). For multi-dimensional problems, when the spatial discretization step sizes are decreased, the number of unknowns of the algebraic equation system arising from the discretization procedure will increase quickly, which leads to rapid increase in computer memory and CPU time. The fractional step method reduces multi-dimensional problems to a series of uncoupled one-dimensional problems, which results in very low execution time and storage. In this paper, we combine a fractional step method with the ELLAM method to develop a fractional step ELLAM (FS-ELLAM) approach for two-dimensional unsteady convection–diffusion problems. In the framework of the fractional step method, we split the two-dimensional convection–diffusion problem into two uncoupled one-dimensional convection–diffusion equations in every time step interval. Then, at each time level, we apply the ELLAM method to solve the one-dimensional subproblems, which simulates accurately the corresponding sub-dimensional equations. The developed FS-ELLAM approach takes the advantages of both the ELLAM method and the fractional step method. It reduces temporal errors and eliminates non-physical oscillation and excessive numerical diffusion. It generates an accurate numerical solution even if large time and coarse spatial step sizes are used. It also treats boundary conditions effectively. Because of the application of the fractional step technique, the FS-ELLAM approach reduces the computation duration and the requirement of large memory in simulation. Numerical experiments show the efficient performance of the developed FS-ELLAM approach. The technique can be easily applied to three-dimensional large-scale convection–diffusion problems and the procedure can be solved by parallel computing systems.

This paper is organized as follows. In Section 2, we introduce the mathematical model and the fractional step technique. Then, we propose the fractional step ELLAM approach for two-dimensional convection–diffusion problems in Section 3. Numerical experiments are given and analysed in detail in Section 4. Finally, some conclusions are addressed in Section 5.