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Energy-diminishing integration of gradient systems. (English) Zbl 1321.65115

Gradient systems model dissipative physical phenomena. A key feature is the monotonic decay of the potential along the trajectories. Numerical methods for the approximation of the trajectories should preserve this monotonicity. High-order methods with good damping properties for very stiff problems are of special interest.
For unconstrained gradient systems in coordinates \(G(y)\dot{y} = - \nabla U(y)\), implicit Runge-Kutta methods (IRK), discrete-gradient methods (DGM) and averaged vector field collocation methods (AVC) are investigated.
While monotonicity of DGM and AVC does not require additional step-size restrictions, this is not the case for IRK.
Concerning stability, roles are exchanged. For simple potentials, DGM, respectively AVC, reduce to the implicit midpoint rule respectively Gauss collocation, both showing no damping for very stiff systems, whereas IRKs like Radau IIA strongly damp eigencomponents belonging to large eigenvalues. Finally, the results above are generalized to constrained gradient systems.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L04 Numerical methods for stiff equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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