Three-term recurrence iterations for energy-based models
Provides a novel numerical method for dissipation-preserving simulation in computational physics and engineering, though the improvement is incremental.
The authors develop three-term recurrence iteration schemes for energy-based models that preserve dissipation, achieving efficient numerical solutions for the biharmonic heat equation and linear poroelasticity.
It is well-known that the midpoint rule preserves the dissipation inequality if applied to a certain class of energy-based models. We introduce an appropriate scaling of the state variables such that the symmetric part of the resulting iteration matrix is guaranteed to be positive definite. This allows the application of three-term iteration schemes such as the methods of Widlund and Rapoport. Special emphasis is put on examples where the symmetric part is block diagonal such that the computations decouple. This then leads to efficient dissipation-preserving numerical schemes as illustrated in two numerical examples, namely the biharmonic heat equation and linear poroelasticity.