Preconditioning in implicit initial-value problem methods on parallel computers

Abstract

Implicit step-by-step methods for numerically solving the initial-value problem {y′=f(y),y(0)=y 0} usually lead to implicit relations of which the Jacobian can be approximated by a matrix of the special formK=I−hM ⊗J, whereM is a matrix characterizing the step-by-step method andJ is the Jacobian off. Similar implicit relations are encountered in discretizing initial-value problems for other types of functional equations such as VIEs, VIDEs and DDEs. Application of (modified) Newton iteration for solving these implicit relations requires the LU-decomposition ofK. Ifs andd are the dimensions ofM andJ, respectively, then this LU-decomposition is anO(s 3 d 3) process, which is extremely costly for large values ofsd. We shall discuss parallel iteration methods for solving the implicit relations that exploit the special form of Jacobian matrixK. Their main characteristic is that each processor is required to compute LU-decompositions of matrices of dimensiond, so that this part of the computational work is reduced by a factors 3. On the other hand, the number of iterations in these parallel iteration methods is usually much larger than in Newton iteration. In this contribution, we will try to reduce the number of iterations by improving the convergence of such parallel iteration methods by means of preconditioning.