Fundamental Systems of Numerical Schemes for Linear Convection-Diffusion
and Their Relationship to Accuracy
A new approach towards the assessment and derivation of numerical methods
for convection dominated problems is presented, based on the comparison
of the fundamental systems of the continuous and discrete operators.
In two or more space dimensions, the dimension of the fundamental
system is infinite, and may be identified with a circle. This
set is referred to as the true fundamental locus. The fundamental
system for a numerical scheme also forms a locus. As a first application,
it is shown that a necessary condition for the uniform convergence of a
numerical scheme is that the discrete locus should contain the true locus,
and it is then shown it is impossible to satisfy this condition with a
finite stencil. Thus, we obtain a simpler proof of Shishkin's result.
It is shown that the distance between the loci is related to the accuracy
of the schemes provided that the loci are sufficiently close. However,
if the loci depart markedly, then the situation is rather more complicated.
Under suitable conditions, we develop an explicit numerical lower bound on
the attainable relative error in terms of the coefficients in the stencil
characterising the scheme and the loci.
AMS-Classification: 65N12, 65N15, 65N30.
Keywords: artificial viscosity, boundary layer, convection-diffusion,
singular perturbation, uniform convergence.
Appeared in: Computing 66 (2001), 199-229.