Component-based derivation of a parallel stiff ODE solver implemented in a cluster of computers
International Journal of Parallel Programming
Component-Based Derivation of a Parallel Stiff ODE Solver Implemented in a Cluster of Computers
International Journal of Parallel Programming
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Digital filters in adaptive time-stepping
ACM Transactions on Mathematical Software (TOMS)
Hybrid (OpenMP and MPI) parallelization of MFIX: a multiphase CFD code for modeling fluidized beds
Proceedings of the 2003 ACM symposium on Applied computing
Adaptive stepsize based on control theory for stochastic differential equations
Journal of Computational and Applied Mathematics
Fourth-Order Runge---Kutta Schemes for Fluid Mechanics Applications
Journal of Scientific Computing
Adaptive time-stepping and computational stability
Journal of Computational and Applied Mathematics - Special issue: International workshop on the technological aspects of mathematics
Evaluating numerical ODE/DAE methods, algorithms and software
Journal of Computational and Applied Mathematics - Special issue: International workshop on the technological aspects of mathematics
Time-step selection algorithms: adaptivity, control, and signal processing
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
Time-step selection algorithms: Adaptivity, control, and signal processing
Applied Numerical Mathematics
Adaptive time-stepping and computational stability
Journal of Computational and Applied Mathematics - Special issue: International workshop on the technological aspects of mathematics
Evaluating numerical ODE/DAE methods, algorithms and software
Journal of Computational and Applied Mathematics - Special issue: International workshop on the technological aspects of mathematics
Dynamic implicit 3D adaptive mesh refinement for non-equilibrium radiation diffusion
Journal of Computational Physics
Stabilized explicit Runge-Kutta methods for multi-asset American options
Computers & Mathematics with Applications
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In the numerical solution of ODEs by implicit time-stepping methods, a system of (nonlinear) equations has to be solved each step. It is common practice to use fixed-point iterations or, in the stiff case, some modified Newton iteration. The convergence rate of such methods depends on the stepsize. Similarly, a stepsize change may force a refactorization of the iteration matrix in the Newton solver. This paper develops new strategies for handling the iterative solution of nonlinear equations in ODE solvers. These include automatic switching between fixed-point and Newton iterations, investigating the "optimal" convergence rate with respect to total work per unit step, a strategy for when to reevaluate the Jacobian, a strategy for when to refactorize the iteration matrix, coordination with stepsize control. Examples will be given, showing that the new overall strategy works efficiently. In particular, the new strategy admits a restrained stepsize variation without refactorizations, thus permitting the use of a smoother stepsize sequence. The strategy is of equal importance for Runge--Kutta and multistep methods.