Research Area S

Simulation

Coordination: Prof. Dr. S. Le Borne

Efficient and reliable implementations of simulation processes require a thorough understanding of techniques combining adaptive discretization & approximation efforts together with advanced solution procedures on the one hand, and on the other hand, of current and future high-performance computing (HPC) aspects.
The projects in this research area develop novel adaption strategies for kernel-based approximations (S1), simulation strategies for large-scale inverse problems (S3), and develop tailored preconditioning techniques for large-scale fluid dynamic applications (S2).
An efficient use of these technologies usually requires not only a mere re-editing of software parts, but necessitates re-thinking of the employed mathematical approaches and algorithms.

PhD-Projects:

S1 (Adaptive Kernels) - Adaptive kernel-based approaches for fluid flow simulations

S2 (Preconditioning) - Preconditioners for RBF-FD discretized fluid flow problems

S2.2 (Preconditioning) - Low-rank correction for preconditioners in fluid flow problems

S3 (Blood Flow) - Blood Flow modeling and estimation by magnetic particle imaging

S4 (Structure-preserving discretization) - Preserving structures of transport-dominated phenomena discretely

S1: Adaptive Kernels

Principal Investigator: Armin Iske

PhD student: Juliane Entzian

Adaptive particle methods relying on meshfree kernels (e.g. radial basis functions) are popular tools for the numerical solution of partial differential equations [Fasshauer & McCourt (2015), Griebel & Schweitzer (2017)]. In particular, the utility of smoothed particle hydrodynamic (SPH) methods [Casulli (1990),Monaghan (2005), Vila (1999)] has been demonstrated for fluid flow simulations in the geosciences [Brecht et al. (2017),Fornberg & Flyer (2015), Iske & Randen (2005)] and other areas, including e.g. (mathematical) data analysis, and kernel-based (machine) learning. The application of adaptive kernel-based particle methods requires particular care, especially for their critical interaction with high order flux evaluations [Ben-Artzi & Falcovitz (2003),Goetz & Iske (2016),Toro (2009)]. The objective of this project is the development of novel concepts for the applicationoriented design of adaptive kernel methods to
satisfy the ever increasing demands of large-scale fluid dynamic computations [Ferrari et al. (2009)]. The innovation of this research refers to recent advances in the design of non-standard kernels providing highly flexible approximations, as they are required in contemporary applications of mathematical data science.

S2: Preconditioning

Principal Investigator: Sabine Le Borne

PhD student: Rebekka Beddig

We are interested in the numerical simulation of buoyancy-driven fluid flows described by the Boussinesq approximation which combines the Navier-Stokes equations enhanced with a Coriolis term and a temperature advection-diffusion equation. The discretization and linearization lead to a sequence of linear systems of equations of saddle point form. The goal is to enhance existing preconditioners from the literature by multiplicative low-rank correction schemes and to assess the quality of such low-rank updates.

S2.2: Preconditioning

Principal Investigator: Sabine Le Borne

Recently, a new meshfree discretization technique for partial differential equations (PDEs) based on radial basis functions (RBF) has been introduced [Fornberg & Flyer (2015)]. Given a set of scattered nodes, RBF-interpolation is used to compute a (local) differentiation stencil (RBF-FD), leading to a sparse linear system of equations. The novelty and objective of this project is the development and analysis of rank-structured preconditioners for the linear systems of equations resulting from meshfree RBF-FD discretization of fluid flow problems.

S3: Blood Flow

Principal Investigator: Christina Brandt

PhD student: Stephanie Blanke

In cardiovascular diagnostics, magnetic particle imaging (MPI) has been developed to acquire blood flow patterns inside arteries. This new imaging modality offers benefits step and of the blood flow which requires efficient minimization algorithms for real-time computations. The problem will advance existing approaches for combined image and flow reconstruction dedicated to denoising and motion estimation to real-time image reconstruction and flow estimation for MPI.

S4: Structure-preserving discretization

Principal Investigator: Hendrik Ranocha

Discontinuous Galerkin (DG) methods are increasingly popular numerical schemes for transport-dominated phenomena such as conservation laws describing the airflow around a plane or car and the evolution of water surfaces, e.g., in rivers and the ocean. In recent years, connections to finite difference methods based on so-called summation by parts (SBP) operators have been discovered. SBP operators mimic integration by parts discretely and enable the development of structure-preserving numerical methods, e.g., entropy-stable schemes and shock-capturing techniques. For many realistic applications, DG methods based on SBP operators need to be coupled to invariant domain preserving techniques, e.g., to guarantee to positivity of the water height in a flooding simulation. The objective of this project is the development of new computational techniques in this realm to increase robustness and computational efficiency. Research topics include the design of SBP operators tailored to specific needs, their application in innovative semidiscretizations for transport-dominated phenomena, the development of optimized time integration methods, and efficient approaches for adaptivity in space and time. The novel methods will be implemented in open source software such as the Julia package Trixi.jl for conservation laws.