Thomas F. Russell

Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference Procedures

Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form

An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation

cu_t + bu_x - (au_x )_x = f

Time Stepping Along Characteristics with Incomplete Iteration for a Galerkin Approximation of Miscible Displacement in Porous Media

. Optimal order error estimates in

An overview of research on Eulerian–Lagrangian localized adjoint methods (ELLAM)

L^2

Some improved error estimates for the modified method of characteristics

and

A finite‐volume Eulerian‐Lagrangian Localized Adjoint Method for solution of the advection‐dispersion equation

W^{1,2}

Efficient Time-Stepping Methods for Miscible Displacement Problems in Porous Media

are derived for the finite element procedure. Various error estimates are presented for a variety of finite difference methods. The estimates show that, for convection-dominated problems

Eulerian-Lagrangian Localized Adjoint Methods for a Nonlinear Advection-Diffusion Equation

(b \gg a)

Solution of the advection-dispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method

, these schemes have much smaller time-truncation errors than those of standard methods. Extensions to n-space variables and time-dependent or nonlinear coefficients are indicated, along with applications of the concepts to certain problems described by systems of differential equations.

Shape Functions for Velocity Interpolation in General Hexahedral Cells

Abstract Many numerical methods use characteristic analysis to accommodate the advective component of transport. Such characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. A generalization of characteristic methods can be developed using an approach that we refer to as an Eulerian-Lagrangian localized adjoint method (ELLAM). This approach is a space-time extension of the optimal test function (OTF) method. The method provides a consistent formulation by defining test functions as specific solutions of the localized homogeneous adjoint equation. All relevant boundary terms arise naturally in the ELLAM formulation, and a systematic and complete treatment of boundary condition implementation results. This turns out to have significant implications for the calculation of boundary fluxes. An analysis of global mass conservation leads to the final ELLAM approximation, which is shown to possess the conservative property. Numerical calculations demonstrate the behaviour of the method with emphasis on treatment of boundary conditions. Discussion of the method includes ideas on extensions to higher spatial dimensions, reactive transport, and variable coefficient equations.