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

TLDR

The authors combine the method of characteristics with finite element and finite difference schemes to solve the convection‑dominated parabolic equation \(cu_t+bu_x-(a u_x)_x=f\), presenting error estimates and extensions to multi‑dimensional, time‑dependent, or nonlinear settings. They prove optimal \(L^2\) and \(W^{1,2}\) error bounds for the finite element approach and demonstrate that, for \(b\gg a\), the resulting schemes achieve substantially lower time‑truncation errors than conventional methods.

Abstract

Finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form $cu_t + bu_x - (au_x )_x = f$. Optimal order error estimates in $L^2 $ and $W^{1,2} $ 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 $(b \gg a)$, 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.