Exponential Integrators: From Implementations to Order Reduction, Simulation of incompressible particle-based fluid flow inspired by optimal transport

Abstract: Exponential Integrators: From Implementations to Order Reduction
Exponential integrators are a class of numerical methods designed for the efficient integration of evolution equations with stiff linear parts. By treating the linear operator through matrix exponentials, these methods provide an attractive approach to problems arising from the spatial discretization of partial differential equations. We provide an overview of exponential integrators, with an emphasis on efficient implementation and applications to evolution equations involving unbounded operators. We introduce the basic principles of exponential Runge--Kutta methods, together with a brief overview of several approaches for computing matrix exponentials and $\varphi$-function actions, including Krylov subspace, Leja interpolation, and Chebyshev approximation methods. In this context, we compare these approximation methods with explicit Runge--Kutta methods of orders two and four for computing the action of the exponential and related matrix functions. A performance model is established by accounting for the cost of the different operations required to implement the considered algorithms. The results show that, depending on the specific setting, exponential integrators either outperform or perform similarly to explicit Runge--Kutta schemes. We also discuss order reduction in exponential Runge--Kutta methods for problems involving unbounded and non-commuting operators. The mechanisms leading to order reduction are discussed. The analysis shows that first- and second-order methods retain their expected order under mild regularity conditions on the initial data $u_0$, whereas third- and fourth-order methods can exhibit order reduction. In particular, for third-order methods, the analysis identifies schemes for which order reduction can be essentially eliminated under suitable regularity assumptions. Numerical experiments illustrating the theoretical results are provided. The talk concludes with perspectives on the analysis and efficient implementation of exponential integrators for stiff evolution equations and partial differential equations.

Simulation of incompressible particle-based fluid flow inspired by optimal transport
The simulation of particle-based fluid flow is usually compressible due to non-constant density, finite speed of sound from interaction effects, and lack of a global pressure field. By combining results from optimal transport theory, especially a globally optimal assignment problem solver with a formulation corresponding to an incompressible particle-based fluid flow, the incompressible Euler flow was first approximated in a new computational formulation. After inclusion of a viscosity step, known qualitative solutions for the Navier-Stokes equation can be demonstrated, such as for a Karman vortex street. These showed the initial suitability of the method for solving inviscid and viscous flow problems and hopefully, after successful validation, the introduced solver will be efficient for such problems as homogenization flows and mixtures of different density fluids.

Computer scienceMathematics

Audience: researchers in the topic

Modelling of materials - theory, model reduction and efficient numerical methods (UNCE MathMAC)