Blended implementation of block implicit methods for ODEs
Applied Numerical Mathematics
The BiM code for the numerical solution of ODEs
Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
Speeding up Netwton-type iterations for stiff problems
Journal of Computational and Applied Mathematics
Optimizing locality and scalability of embedded Runge--Kutta solvers using block-based pipelining
Journal of Parallel and Distributed Computing
Journal of Computational and Applied Mathematics
Recent advances in linear analysis of convergence for splittings for solving ODE problems
Applied Numerical Mathematics
An iterated Radau method for time-dependent PDEs
Journal of Computational and Applied Mathematics
Speeding up Newton-type iterations for stiff problems
Journal of Computational and Applied Mathematics
Efficient iterations for Gauss methods on second-order problems
Journal of Computational and Applied Mathematics
Blended implicit methods for the numerical solution of DAE problems
Journal of Computational and Applied Mathematics
Efficient implementation of Gauss collocation and Hamiltonian boundary value methods
Numerical Algorithms
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It often happens that iteration processes used for solving the implicit relations arising in ODE-IVP methods only start to converge rapidly after a certain number of iterations. Fast convergence right from the beginning is particularly important if we want to use so-called step-parallel iteration in which the iteration method is concurrently applied at a number of step points. In this paper, we construct highly parallel iteration methods that do converge fast from the first iteration on. Our starting point is the PDIRK method (parallel, diagonally implicit, iterated Runge--Kutta method), designed for solving implicit Runge--Kutta equations on parallel computers. The PDIRK method may be considered as a Newton-type iteration in which the Newton Jacobian is "simplified" to block-diagonal form. However, when applied in a step-parallel mode, it turns out that its relatively slow convergence, or even divergent behavior, reduces the effectiveness of the step-parallel scheme. By replacing the block-diagonal Newton Jacobian approximation in PDIRK by a block-triangular approximation, we do achieve convergence right from the beginning at a modest increase of the computational costs. Our convergence analysis of the block-triangular approach will be given for the wide class of general linear methods, but the derivation of iteration schemes is limited to Runge--Kutta-based methods. A number of experiments show that the new parallel, triangularly implicit, iterated Runge--Kutta method (PTIRK method) is a considerable improvement over the PDIRK method.