Efficient high-order structure-preserving numerical methods for nonlinear evolution equations

• Applicant: Professor Dr. Hendrik Ranocha
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 513301895

Many problems in science and engineering are modeled by evolution equations. Some examples of such nonlinear partial differential equations include large-scale astrophysical plasmas, the airflow around a car, water waves, biological systems, or semiconductor devices. Despite their broad range of applications, evolution equations often share some common structures. Indeed, the qualitative behavior of such systems is often determined by how the total energy or entropy changes in time. General entropy methods have also proven successful in the mathematical theory of evolution equations. Moreover, preserving such entropy structures discretely usually results in robust algorithms, improved qualitative properties, a correct asymptotic behavior, and reduced numerical errors. We will develop and analyze high-order structure-preserving numerical methods for a range of evolution equations, including compressible fluid dynamics, Hamiltonian systems, nonlinear wave equations, and other nonlinear equations with higher-order dispersive and/or dissipative terms. While the focus is on efficient time integration methods preserving the correct evolution of an entropy functional, spatial discretizations are also considered. In particular, we will build on our recent works on relaxation methods in time and summation by parts (SBP) operators in space, combine them, and extend them significantly. Efficient implementations of all methods using the modern programming language Julia will be published in open source packages. These are anticipated to result in significant gains of efficiency, robustness, and accuracy compared to existing technology. Hence, they enable the application of numerical methods to problems in science and engineering that have been intractable before. Moreover, we will advance the understanding of structure-preserving numerical methods, e.g., by analyzing the error-growth rates.

• DFG Programme: Research Grants
• International Connection: Saudi Arabia
• Cooperation Partner: Professor David Ketcheson