## Final Report Summary - STIMULUS (Space-Time Methods for Multi-Fluid Problems on Unstructured Meshes)

The main objective of the STiMulUs project was the development of new classes of high order (better than second order) accurate shock capturing numerical methods for the solution of nonlinear systems of hyperbolic partial differential equations (PDE) with stiff source terms and non-conservative products on fixed and moving unstructured meshes. These mathematical equations usually represent the fundamental physical principles of the conservation of mass, momentum and energy. Typical examples for such PDE are the Euler equations for compressible fluid dynamics or the magneto-hydrodynamics equations for plasma flow. Usually, the solutions of the above-mentioned PDE contain at the same time discontinuities (shock waves) and smooth features, like sound waves and vortices. Resolving both types of features with the same numerical scheme is notoriously difficult, in particular on unstructured meshes which are convenient for the discretization of complex geometries that appear in many areas of science and engineering. A particularly challenging application area is given by inertial confinement fusion (ICF) flows, involving huge pressure, temperature and density jumps and thus very strong shock waves and high compression, but at the same time these flows also exhibit a series of well-known hydrodynamic instabilities like the Kelvin-Helmholtz instability, the Rayeigh-Taylor and the Richtmyer-Meshkov instability, which are believed to be the main cause why previous ICF experiments were less successful than theoretically predicted.

In order to investigate ICF flows, the STiMulUs project has developed the first arbitrary high order accurate Lagrangian finite volume and discontinuous Galerkin finite element schemes on moving unstructured meshes in 2D and 3D that are able to resolve very strong shock waves and small scale flow instabilities at the same time. On moving meshes there may appear highly compressed and very tiny cells, which typically restrict the maximum time step of the simulation. To circumvent this problem, we have developed the first family of Lagrangian schemes with time-accurate local time stepping (LTS), where each element is allowed to run with its own optimal time step, but without sacrificing accuracy and conservation. It was also possible to show that the hydrodynamic instabilities arising in ICF flows can be successfully reduced or even suppressed at the aid of external magnetic fields. These achievements can be seen as the first major outcome of the project.

To further study hydrodynamic instabilities and shock waves with great precision we have developed a radically different and totally new nonlinear shock capturing technique for dealing with shock waves in discontinuous Galerkin finite element schemes on unstructured meshes as well as on space-time adaptive grids. Our new approach uses a novel a posteriori concept for the detection of invalid numerical solutions and is even able to cure a certain set of numerical errors that occur during run time (floating point errors, unphysical negative densities or pressures). This greatly enhances the robustness of our numerical schemes. The numerical solution in detected troubled cells is discarded and recomputed with a more robust scheme on a fine subgrid. Computational experiments have been carried out on modern supercomputers with several thousands of CPU cores and with numerical schemes up to tenth order of accuracy in space and time for problems involving at the same time shocks and smooth flow features, using more than one billion degrees of freedom to represent the discrete solution. This can be seen as the second major achievement of the project.

During our research, we have also made an occasional finding that allowed us to develop a completely new family of numerical methods that did not exist before: discontinuous finite elements on staggered meshes, where the pressure is defined on the cells of a main grid and the velocity field is defined on a face-based staggered dual mesh. While the benefits of staggered grids are well-known in the framework of finite difference and finite volume schemes, in the field of finite element schemes this approach is essentially unknown. The method has several interesting properties that we were also able to prove mathematically. During the STiMulUs project we were able to develop and apply this new family of schemes to a set of well-known equations that are very important for practical applications, namely the shallow water equations for free surface flows, the incompressible Navier-Stokes equations and the full compressible Navier-Stokes equations in two and three space dimensions. In the last case, the new method is able to describe the entire range of Mach numbers, which is the ratio between flow speed and sound speed, including smooth nearly incompressible flows at low Mach numbers, as well as high Mach number flows including shock waves. It has to be underlined that it is very difficult to construct numerical methods that work well for both, very low and very high Mach numbers at the same time. Having developed such a scheme is the third major achievement of the STiMulUs project.

Last but not least, we have carried out a substantial amount of research concerning the choice of suitable mathematical models for the description of non-ideal fluid and plasma flows as well as multi-material flows. In this context we have found a unified model of continuum mechanics that has been proposed by Godunov & Romenski and Peshkov & Romenski. The governing PDE system is able to describe at the same time the behavior of nonlinear elastic and elasto-plastic solids, non-ideal liquids as well as gases, and all these media can be coupled with the effects of electro-magnetic fields. In other words, the above-mentioned mathematical model describes the entire range of continuum physics from solids over fluids to electro-magnetic waves in one single system of partial differential equations. Further to that, it is a provably first order symmetric hyperbolic and thermodynamically compatible system with a very elegant mathematical structure that can be directly derived from variational principles. All present dissipative processes are described via stiff algebraic relaxation source terms instead of conventional parabolic terms. Hence, the numerical methods developed in this ERC project are particularly suitable for the solution of this model. During the STiMulUs project we were able to provide an asymptotic analysis of the Godunov-Peshkov-Romenski system in the stiff relaxation limit and were able to solve it for the first time numerically with very high order accurate shock capturing schemes in two and three space dimensions. Numerical experiments have been carried out for non-ideal plasma flows, for nonlinear elasto-plastic solids, as well as for electromagnetic wave propagation in heterogeneous media. This can be seen as the fourth major achievement of the project and will be the basis of further research in this very promising direction.

In order to investigate ICF flows, the STiMulUs project has developed the first arbitrary high order accurate Lagrangian finite volume and discontinuous Galerkin finite element schemes on moving unstructured meshes in 2D and 3D that are able to resolve very strong shock waves and small scale flow instabilities at the same time. On moving meshes there may appear highly compressed and very tiny cells, which typically restrict the maximum time step of the simulation. To circumvent this problem, we have developed the first family of Lagrangian schemes with time-accurate local time stepping (LTS), where each element is allowed to run with its own optimal time step, but without sacrificing accuracy and conservation. It was also possible to show that the hydrodynamic instabilities arising in ICF flows can be successfully reduced or even suppressed at the aid of external magnetic fields. These achievements can be seen as the first major outcome of the project.

To further study hydrodynamic instabilities and shock waves with great precision we have developed a radically different and totally new nonlinear shock capturing technique for dealing with shock waves in discontinuous Galerkin finite element schemes on unstructured meshes as well as on space-time adaptive grids. Our new approach uses a novel a posteriori concept for the detection of invalid numerical solutions and is even able to cure a certain set of numerical errors that occur during run time (floating point errors, unphysical negative densities or pressures). This greatly enhances the robustness of our numerical schemes. The numerical solution in detected troubled cells is discarded and recomputed with a more robust scheme on a fine subgrid. Computational experiments have been carried out on modern supercomputers with several thousands of CPU cores and with numerical schemes up to tenth order of accuracy in space and time for problems involving at the same time shocks and smooth flow features, using more than one billion degrees of freedom to represent the discrete solution. This can be seen as the second major achievement of the project.

During our research, we have also made an occasional finding that allowed us to develop a completely new family of numerical methods that did not exist before: discontinuous finite elements on staggered meshes, where the pressure is defined on the cells of a main grid and the velocity field is defined on a face-based staggered dual mesh. While the benefits of staggered grids are well-known in the framework of finite difference and finite volume schemes, in the field of finite element schemes this approach is essentially unknown. The method has several interesting properties that we were also able to prove mathematically. During the STiMulUs project we were able to develop and apply this new family of schemes to a set of well-known equations that are very important for practical applications, namely the shallow water equations for free surface flows, the incompressible Navier-Stokes equations and the full compressible Navier-Stokes equations in two and three space dimensions. In the last case, the new method is able to describe the entire range of Mach numbers, which is the ratio between flow speed and sound speed, including smooth nearly incompressible flows at low Mach numbers, as well as high Mach number flows including shock waves. It has to be underlined that it is very difficult to construct numerical methods that work well for both, very low and very high Mach numbers at the same time. Having developed such a scheme is the third major achievement of the STiMulUs project.

Last but not least, we have carried out a substantial amount of research concerning the choice of suitable mathematical models for the description of non-ideal fluid and plasma flows as well as multi-material flows. In this context we have found a unified model of continuum mechanics that has been proposed by Godunov & Romenski and Peshkov & Romenski. The governing PDE system is able to describe at the same time the behavior of nonlinear elastic and elasto-plastic solids, non-ideal liquids as well as gases, and all these media can be coupled with the effects of electro-magnetic fields. In other words, the above-mentioned mathematical model describes the entire range of continuum physics from solids over fluids to electro-magnetic waves in one single system of partial differential equations. Further to that, it is a provably first order symmetric hyperbolic and thermodynamically compatible system with a very elegant mathematical structure that can be directly derived from variational principles. All present dissipative processes are described via stiff algebraic relaxation source terms instead of conventional parabolic terms. Hence, the numerical methods developed in this ERC project are particularly suitable for the solution of this model. During the STiMulUs project we were able to provide an asymptotic analysis of the Godunov-Peshkov-Romenski system in the stiff relaxation limit and were able to solve it for the first time numerically with very high order accurate shock capturing schemes in two and three space dimensions. Numerical experiments have been carried out for non-ideal plasma flows, for nonlinear elasto-plastic solids, as well as for electromagnetic wave propagation in heterogeneous media. This can be seen as the fourth major achievement of the project and will be the basis of further research in this very promising direction.