Convection-Diffusion Problems: An Introduction to Their Analysis and Numerical Solution

This book was written for those having some familiarity with numerical methods and their analysis who wish to investigate convection-diffusion problems. These have recently gotten a good deal of attention in the research literature, but standard texts are very challenging for those without previous experience with convection-diffusion and associated singularly perturbed differential equations.

 

It is still not an easy book. Some background with two-point boundary value problems and their numerical solution is sufficient for the early chapters. Later, familiarity with partial differential equations, Lebesgue \( L^{p} \) and Sobolev \( H^{k} \) spaces is necessary. For the later chapters, a general understanding of finite element methods is also expected.

 

Although convection-diffusion equations arise in several situations, the most common source of problems comes from linearizations of Navier-Stokes equations, particularly those with large Reynolds number. (Big Reynolds numbers are associated with more turbulent fluid motion; low numbers are characteristic of the much simpler laminar flow.) Other convection-diffusion problems arise in financial modeling (the Black-Scholes equation), semiconductor physics (drift-diffusion equations), and Brownian motion (the Fokker-Planck equation).

 

Convection-diffusion equations are elliptic partial differential equations where the second-order derivatives represent the diffusion and the first-order derivatives model the convection or transport processes. Diffusion is usually the dominant term in classical boundary layer problems, and convection has a limited influence on the analysis. When diffusion dominates, the analysis is simpler and standard numerical solution methods perform well. When convection dominates, standard methods become unstable and special techniques are needed to approximate solutions numerically. This book mostly concerns itself with the convection-dominated regime.

 

The treatment of convection-dominated problems begins by setting up a singular perturbation of the basic equation. This means modifying the convection-diffusion equation by multiplying the second order derivatives that represent diffusion by a parameter \( \epsilon \) where \( 0 < ε \leq 1 \). As \( \epsilon \) approaches zero, the equation becomes more and more dominated by convection. Robust numerical methods in this regime require a priori bounds on derivatives of solutions. Without these, the derivatives of apparent solutions oscillate wildly.

 

After setting up the preliminaries and filling in background material, the authors consider asymptotic solutions to convection-diffusion equations in one dimension, with emphasis on the behavior of the derivatives of these solutions. Much of the remainder of the book concentrates on numerical methods. This begins with an application of finite difference methods in one dimension and then moves on to two dimensions and finite difference methods there. The final and longest chapter concentrates on finite element methods. The authors note that the usual Galerkin finite element method with an equidistant mesh leads to a computed solution with large oscillations. To stabilize the computation a special choice of test functions, a special mesh, or a combination of both is needed. The authors devote a good deal of attention to these questions.

 

This is a well-written text best suited to readers who are interested in pursuing challenging problems in convection-diffusion and who have at least a modest background in partial differential equations and their numerical solution.

---

 

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films. He did his PhD work in dynamical systems and celestial mechanics.