Multirate numerical integration for ordinary differential equations

For large systems of ordinary differential equations (ODEs), some components may show a more active behavior than others. To solve such problems nu- merically, multirate integration methods can be very efficient. These methods enable the use of large time steps for slowly varying components and small steps for rapidly varying ones. In this thesis we design, analyze and test multirate methods for the numerical solution of ODEs. A self-adjusting multirate time stepping strategy is presented in Chapter 1. In this strategy the step size for a particular system component is determined by the local temporal variation of this solution component, in contrast to the use of a single step size for the whole set of components as in the traditional methods. The partitioning into different levels of slow to fast components is performed automatically during the time integration. The number of activity levels, as well as the component partitioning, can change in time. Numerical experiments confirm that with our strategy the efficiency of time integration methods can be significantly improved by using large time steps for inactive components, without sacrificing accuracy. A multirate scheme, consisting of the ¿-method with one level of temporal local refinement, is analysed in Chapter 2. Missing component values, required during the refinement step, are computed using linear or quadratic interpolation. This interpolation turns out to be important for the stability of the multirate scheme. Moreover, the analysis shows that the use of linear interpolation can lead to an order reduction for stiff systems. The theoretical results are confirmed in numerical experiments. Two multirate strategies, recursive refinement and the compound step strat- egy are compared in Chapter 3. The recursive refinement strategy has somewhat larger asymptotic stability regions than the compound step strategy. The com- pound step strategy, by avoiding the extra work of doing the macro step for all the components, looses some stability properties compared to the recursive re- finement strategy. It can also lead to more complex algebraic implicit systems, which are difficult to solve numerically. The construction of higher-order multirate Rosenbrock methods is discussed in Chapter 4. Improper treatment of stiff source terms and use of lower-order interpolants can lead to an order reduction, where we obtain a lower order of consistency than for non-stiff problems. We recommend a strategy of avoidance of the order reduction for problems with a stiff source term. A multirate method based on the fourth-order Rosenbrock method RODAS and its third-order dense output has been designed. This multirate RODAS method has shown very good results in numerical experiments, and it is clearly more efficient than other considered multirate methods in these tests. Explicit multirate and partitioned Runge-Kutta schemes for semi-discrete hyperbolic conservation laws are analysed in Chapter 5. It appears that, for the considered class of multirate methods, it is not possible to construct a multirate scheme which is both locally consistent and mass-conservative. The analysis shows that, in spite of local inconsistencies, global convergence is still possible in all grid points