Structure preserving approximation of dissipative evolution problems
For researchers in numerical analysis and scientific computing, this provides a unified framework for structure-preserving discretization of dissipative systems, though it is an incremental extension of known variational and Galerkin techniques.
The paper presents a general framework for numerically approximating dissipative evolution problems while preserving their energy/entropy structure. The approach uses Galerkin methods in space and discontinuous Galerkin in time to automatically inherit dissipative behavior, demonstrated on examples from PDEs to Hamiltonian systems.
We present a general abstract framework for the systematic numerical approximation of dissipative evolution problems. The approach is based on rewriting the evolution problem in a particular form that complies with an underlying energy or entropy structure. Based on the variational characterization of smooth solutions, we are then able to show that the approximation by Galerkin methods in space and discontinuous Galerkin methods in time automatically leads to numerical schemes that inherit the dissipative behavior of the evolution problem. The proposed framework is rather general and can be applied to a wide range of applications. This is demonstrated by a detailed discussion of a variety examples ranging from diffusive partial differential equations to Hamiltonian and gradient systems.