Computational Stability and Adaptive Strategies - An Experimental Study of ODE Software

• The algorithmic content of adaptive ODE/DAE software is dominated by a considerable amount of control structures, support algorithms and logic. The control logic has been largely heuristic and lacking support by analysis. In this thesis we study adaptive time-stepping based on control theory, combined with other carefully selected control strategies. Thus we argue that ODE/DAE software can be constructed and analyzed by proven, "standard" scientific techniques instead of heuristics. We show that carefully designed adaptive algorithms have a most significant impact on the reliability and computational stability of ODE/DAE codes. The latter notion implies that minor changes of the computational setup must only lead to small changes in the computed result: the software must be computationally well-conditioned. A series of computational experiments with the standard implementations of DASSL and RADAU5 are compared to results obtained with modified versions of these codes, where the improved strategies build on the theory mentioned above. These include stepsize control based on digital filters. The experiments demonstrate that these "minor" algorithmic changes strongly improve computational stability at no extra computational expense.

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

algorithm analysis

Mathematical software

RADAU5

DASSL

digital filters

PI control

adaptive time-stepping

stepsize control

computational stability

test protocol

Computer science

numerical analysis

systems

control

Datalogi

numerisk analys

system

kontroll

Publication and Content Type

dok (subject category)

vet (subject category)