Mohamed Al-Lawatia

Second-order characteristic methods for advection-diffusion equations and comparison to other schemes

Abstract We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order Runge–Kutta approximation of the characteristics within the framework of the Eulerian–Lagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive definite for most combinations of the boundary conditions. Extensive numerical experiments are presented which compare the performance of these two Runge–Kutta methods to many other well perceived and widely used methods which include many Galerkin methods and high resolution methods from fluid dynamics.

A characteristic domain decomposition and space–time local refinement method for first-order linear hyperbolic equations with interfaces

We develop a characteristic-based domain decomposition and space{time local refinement method for firstorder linear hyperbolic equations. The method naturally incorporates various physical and numerical interfaces into its formulation and generates accurate numerical solutions even if large time-steps are used. The method fully utilizes the transient and strongly local behavior of the solutions of hyperbolic equations and provides solutions with significantly improved accuracy and efficiency. Several numerical experiments are presented to illustrate the performance of the method and for comparison with other domain decomposition and local refinement schemes. c 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 1{28, 1999

An Eulerian-Lagrangian Solution Technique for Single-Phase Compositional Flow in Three-Dimensional Porous Media

We develop an Eulerian-Lagrangian iterative solution technique for accurate and efficient numerical simulation of single-phase compositional flow through three-dimensional compressible porous media in the presence of multiple injection and production wells. In the numerical simulator, an Eulerian-Lagrangian method is used to solve the convection-diffusion transport equations and a mixed finite-element method is used to solve the pressure equation. Numerical experiments are presented to investigate the performance of the method. Moreover, because of the Lagrangian nature of the algorithm, the simulator is capable of using large time steps and coarse spatial grids to generate accurate and stable results.

A characteristic method for porous medium flow

We present an efficient and accurate characteristic method for the solution of the unsteady-state advection-diffusion transport equations, which arise in the mathematical models for flow and transport processes in porous media. The advantages of this method include symmetrising the governing transport equation, treating boundary conditions naturally in the formulation, and conserving mass. Moreover, because of the Lagrangian nature of the algorithm used, the method generates accurate numerical solutions even when large time steps and coarse spatial grids are used in the simulation. Numerical experiments are presented to illustrate the performance of the method.

A Comparison of ELLAM with ENO/WENO Schemes for Linear Transport Equations

We present an Eulerian-Lagrangian localized adjoint method (EL-LAM) for linear advection-reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to compare the performance of the ELLAM scheme with the essentially non-oscillatory (ENO) schemes and weighted essentially non-oscillatory (WENO) schemes, which shows that the ELLAM scheme outperforms ENO and WENO schemes in the context of linear transport PDEs.

Restoring oceanic islands' ecosystems by controlling introduced predator species

Abstract In this paper, we analyze a predator—prey model of oceanic islands with switching strategy of opportunistic superpredators. The essential mathematical features are analyzed with the help of equilibrium local and global stability analysis. We find seven equilibrium possibilities and establish conditions under which the three species, prey (birds), mesopredator (rabbits) and superpredator (cats), coexist. Moreover, we discuss ways to save the bird population. Experimental data simulation and brief discussion conclude the paper.