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Parallel ODE-solvers with stepsize control. (English) Zbl 0707.65051
The present paper continues a recent investigation by the authors [J. Comput. Appl. Math. 25, 341-350 (1989; Zbl 0675.65134)] on parallel implementation of one-step methods \(y_{n+1}=F_{n+1}(y_ n)\), \(n=0,...,N-1\), for ordinary differential equations (ODEs), where the trajectory \((y_ 0,y_ 1,...,y_ N)\) is iteratively computed as the fixed point of the transformation \(\phi (u_ 0,u_ 1,...,u_ N)=(u_ 0,F_ 1(u_ 0),...,F_ N(u_{N-1}))\). Availing of p processors the block \((u^ k_ s,...,u^ k_{s+p})\) initialized as \((u^ 0_ 0:=y_ 0,...,u^ 0_ p)\) is handled concurrently each iteration and shifts forward at least one index each iteration.
A moving mesh method based on some mechanism of self-adapting stepsize is developed. The strategy consists in performing the stepsize control by changing the mesh after each iteration. The practical implementation is explained in detail, the attainable speedup is studied and numerical examples illustrate the method.
Reviewer: R.Scherer

MSC:
65L05 Numerical methods for initial value problems
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65Y05 Parallel numerical computation
34A34 Nonlinear ordinary differential equations and systems, general theory
Software:
dverk
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References:
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[2] Bellen, A.; Zennaro, M., Parallel algorithms for initial-value problems for difference and differential equations, J. comput. appl. math., 25, 3, 341-350, (1989) · Zbl 0675.65134
[3] Bellen, A.; Zennaro, M.; Vermiglio, R., Parallel ODE solvers with step-size control, Quaderno N. 183, (1989), Dipartimento di Scienze Matematiche, Universita‘di Trieste · Zbl 0707.65051
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