Stiff and singular problems

Abstract

This chapter discusses some techniques for handling stiff and singular problems, using Chebyshev methods. The solution of a stiff problem is regular but exhibits large variations in a region of small extent. For convergence reasons, the collocation points cannot be clustered arbitrarily in the rapid variation region, so that an appropriate distribution of the collocation points must be obtained through a coordinate transformation, preferably self-adaptive. Another way is to use a domain decomposition method which can be combined with the preceding one. The presentation of these various approaches is the subject of the first part of this chapter. The second part is devoted to the calculation of singular solutions, which have only a very small number of bounded derivatives. For such functions, the rate of convergence of the Chebyshev approximation is only algebraic and, in the case of a strong singularity, the polynomial representation is subject to oscillations. Two ways to recover high accuracy will be discussed. The first way consists of using a domain decomposition method so that the singularity is shifted to a corner of a subdomain. The second way, which is more efficient, consists of subtracting the most singular part of the solution, so that the computed solution is less singular and can be accurately represented by the Chebyshev approximation.