Less is More: Efficient Numerics for Model Physics Problems in Simple Geometries
CSGF Annual Conference, Washington, D.C., June 23, 2005
Exploring the behavior of model problems is a crucial first step towards understanding any scientific or engineering phenomenon. A key feature of many model problems is a simple geometry. In these cases, high-order numerical methods are extremely powerful because (1) they make it possible to obtain accurate solutions with very few grid points and (2) they are fairly easy to implement. In other words, they make efficient use of computational resources AND programmer time. An important consequence of the computational efficiency afforded by high-order methods is that significant progress can made within scientific computing environments such as MATLAB (which is not usually considered high-performance); the higher-order methods make it possible to take advantage of MATLAB's wide array of built-in functions without sacrificing too much performance.
In this talk, we will discuss the practical application of high-order pseudospectral methods to problems in electrochemical transport. The presence of thin-boundary layers and complicated boundary conditions in these systems makes solution of the governing equations computationally expensive using standard low-order schemes. By using pseudospectral methods, we were able to work completely within the MATLAB framework permitting quick visualization and analysis of the results. A few of the practical issues that may be discussed include: (1) dealing with infinite-domains, (2) computing analytical Jacobians for discrete differential operators, and (3) achieving performance through mixed-language programming.