Effective Numerical Computation of Parameter Dependent Problems

Abstract

We analyse parameter dependent differential-algebraic-equations (DAEs)

$$Ad\prime(x,t,p) + b(x,t,p) = 0.$$

For these systems one is interested in the relation between the numerical solutions x and some associated parameters p. The standard approach is to discretise the equations with respect to the parameters and solve the parameter independent equations afterwards. This approach forces a calculation of the differential equations multiple times (for a huge number of parameter values p). This may lead to high computational costs. By using the already computed solutions to calculate the remaining ones and thus exploiting the smoothness of the solution with respect to the parameters, it is possible to save the majority of the computational cost.