Partial differential equations describe the evolution of physical systems in space and time. Realistic models involve the nonlinear coupling of different physical fields, and require their self-consistent solution. Numerical simulation methods involve various space and time discretization approaches which almost certainly lead to sparse nonlinear and linear systems to be solved.
Julia enables efficient composition of partial differential equations (PDEs) solvers through its multiple dispatch, while also offering access to parallel computing and hardware accelerators such as GPUs. The mini-symposium shall discuss various tools and approaches available in Julia which are relevant for PDEs like sparsity handling, automatic differentiation, mesh generation, linear and nonlinear system solvers, parallelization, pre- and post-processing, and visualization. It shall highlight applications from various fields of science and engineering. At the same time, issues like interoperability between different package ecosystems and missing functionality shall be discussed. An open panel will specifically discuss infrastructure needs, identify gaps, and coordinate development efforts to advance the ecosystem.
Julia has gained significant attention and popularity for the solution of partial differential equations (PDEs), which are used to model a variety of phenomena in mathematics, physics, biology, geophysics, economics, mechanical, civil, and aerospace engineering, to name but a few. Julia's multiple dispatch allows the composition of different building blocks to enable numerical simulation pipelines, while allowing the development of all infrastructure in one language and offering multi-threading, distributed memory and easy access to hardware accelerators such as GPUs.
The aim of the mini-symposium is to bring together the Julia community working on tools and packages for solving PDEs, both from the perspective of users and developers, highlighting current practices, issues, common interests, and future directions. Particular interest is given to bridging the gap between practitioners and package developers whose objective is to obtain increased portability, high performance, efficiency, and seamless composition. The packages and infrastructure presented will focus on typical applications, novel or efficient implementations of numerical algorithms, and introductory overviews of the discretization techniques employed.
Some of the discussed tools will include sparsity handling and automatic differentiation (AD) in the context of PDE models. The mini-symposium will contain an overview of the sparse solvers in Julia and their performance. Tools for mesh generation and visualization will also be discussion.
Another goal of the mini-symposium is to increase visibility of packages and development techniques in Julia for PDEs, in an attempt to pool development efforts and inform potential users. Thus, we reserve some time for an open panel session to discuss various packages, identify users' needs, and potential gaps in the infrastructure to bring the community forward as a whole.
We welcome contributions related to numerical methods and discretization techniques for a broader public, including finite difference, finite element, finite volume, spectral methods, and other techniques, such as model order reduction, machine learning, adjoint-based optimization.
Jürgen Fuhrmann, PhD, is deputy head of the Numerical Mathematics and Scientific Computing group at Weierstrass Institute for Applied Analysis and Stochastics, Berlin. His research topics include finite volume methods, numerical simulation in electrochemistry, semiconductors and other fields, and software design and development for partial differential equations. Since 2018, Julia is his main programming language. He regularly teaches Julia based courses on Advanced Topics from Scientific Computing at TU Berlin.
I am a PhD student at the Johannes Gutenberg University Mainz working at the Institute of Mathematics, under the supervision of Hendrik Ranocha. I am mainly interested in high-order methods, entropy stable/conservative schemes and applications towards atmospheric flows. My contributions to Julia are mostly in Trixi.jl, TrixiAtmo.jl and Ariadne.jl.
I am a Humboldt postdoctoral researcher under Professor Hendrik Ranocha in the Numerical Mathematics group at Johannes Gutenberg University, Mainz.
I work with the Julia packages Tenkai.jl and TrixiLW.jl.
Researcher in Computational Cardiology