Periodic Reporting for period 1 - FASTKiT (Fully Adaptive Simulation Tool for Kinetic Theory)
Berichtszeitraum: 2020-04-01 bis 2022-03-31
Kinetic models are omnipresent in a wide range of scientific and engineering applications. They are derived from the evolution of a particle distribution in position-velocity phase space, and appear, for instance, in the modeling of fusion energy reactors and atmospheric reentry of spacecraft, where classical fluid equations are inaccurate. Further technological progress of these applications requires significant advances in modeling and simulation of kinetic models. The underlying kinetic equations pose severe simulation challenges, due to their inherent high-dimensionality and the presence of a wide range of time scales.
The increased dimensionality in velocity directions can be addressed by an extended set of fluid quantities via moment models or the maximum entropy method. To deal with the stiffness of the equations, asymptotic-preserving time discretization methods need to be used. Since both the stiffness and the accuracy of a kinetic model depend on space and time, the design of numerical methods incorporating fully integrated space-time adaptivity is crucial to allow these methods to be efficiently used in real-world applications.
In this action, the applicant will integrate his expertise on moment models with the experience on projective integration schemes available at the host institution, and extend their applicability towards a wide range of kinetic models hereby achieving the following objectives:
- Develop fully space-time adaptive numerical scheme for kinetic models
- Implement software for space-time adaptive solution of kinetic models
- Compute numerical solutions for real-world applications
The results of FASTKiT will constitute a major step forward in the adaptive simulation of kinetic models. FASTKiT will contribute to the development of technologies for next generation reactors and space exploration efforts, in line with Horizon 2020
The increased dimensionality in velocity directions can be addressed by an extended set of fluid quantities via moment models or the maximum entropy method. To deal with the stiffness of the equations, asymptotic-preserving time discretization methods need to be used. Since both the stiffness and the accuracy of a kinetic model depend on space and time, the design of numerical methods incorporating fully integrated space-time adaptivity is crucial to allow these methods to be efficiently used in real-world applications.
In this action, the applicant will integrate his expertise on moment models with the experience on projective integration schemes available at the host institution, and extend their applicability towards a wide range of kinetic models hereby achieving the following objectives:
- Develop fully space-time adaptive numerical scheme for kinetic models
- Implement software for space-time adaptive solution of kinetic models
- Compute numerical solutions for real-world applications
The results of FASTKiT will constitute a major step forward in the adaptive simulation of kinetic models. FASTKiT will contribute to the development of technologies for next generation reactors and space exploration efforts, in line with Horizon 2020
1) we have combined projective integration and kinetic (moment) models to successfully overcome the time step constraint of standard schemes, leading to a speedup of up to 200.
2) we developed a spatially adaptive projective integration scheme based on domain decomposition and showed that a stable integration of the stiff model is possible, leading to a speedup of more than 800.
3) we developed a fully space-time adaptive projective integration scheme using the notation of embedded Runge-Kutta schemes. This is expected to further reduce runtime for time accurate and steady-state simulations.
4) we derived new shallow water moment models allow to represent vertical variations of velocity and benefit from projective integration, leading to increased accuracy of numerical simulations. These showed that these models exhibit non-trivial steady-states and are stable in equilibrium. We successfully applied the models to sediment transport and used projective integration to reduce the runtime by a factor of 55.
Exploitation and dissemination:
The results listed above have been or will be communicated to the research community with the goal of exploitation in future research projects and applications.
1) published: Projective Integration Schemes for Hyperbolic Moment Equations, J. Koellermeier, G. Samaey, Kinet. Relat. Mod., 14(2), 353-387, 2021
2) submitted: Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps, J. Koellermeier, G. Samaey, submitted
3) in preparation: Projective Integration Methods in the Runge-Kutta Framework and the Extension to Adaptivity in Time, J. Koellermeier, G. Samaey, in preparation
4) published: Shallow Water Moment models for bedload transport problems, J. Garres-Díaz, M. J. Castro, J. Koellermeier, T. Morales de Luna, Commun. in Comput. Phys., 30(3), 903-941, 2021;
Equilibrium Stability Analysis of Hyperbolic Shallow Water Moment equations, Q. Huang, J. Koellermeier, W.-A. Yong, Math. Method. Appl. Sci., 2022;
Recent developments in modeling free-surface flows with vertically-resolved velocity profiles using moments, J. Koellermeier, Proceedings of the 26th Congress of Differential Equations and Applications (CEDYA) and 16th Congress of Applied Mathematics (CMA) 2021, Institutional Repository of the Oviedo University, p.247-252 2021;
submitted: Steady States and Well-balanced Schemes for Shallow Water Moment Equations with Topography, J. Koellermeier, E. Pimentel-Garcia, submitted
2) we developed a spatially adaptive projective integration scheme based on domain decomposition and showed that a stable integration of the stiff model is possible, leading to a speedup of more than 800.
3) we developed a fully space-time adaptive projective integration scheme using the notation of embedded Runge-Kutta schemes. This is expected to further reduce runtime for time accurate and steady-state simulations.
4) we derived new shallow water moment models allow to represent vertical variations of velocity and benefit from projective integration, leading to increased accuracy of numerical simulations. These showed that these models exhibit non-trivial steady-states and are stable in equilibrium. We successfully applied the models to sediment transport and used projective integration to reduce the runtime by a factor of 55.
Exploitation and dissemination:
The results listed above have been or will be communicated to the research community with the goal of exploitation in future research projects and applications.
1) published: Projective Integration Schemes for Hyperbolic Moment Equations, J. Koellermeier, G. Samaey, Kinet. Relat. Mod., 14(2), 353-387, 2021
2) submitted: Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps, J. Koellermeier, G. Samaey, submitted
3) in preparation: Projective Integration Methods in the Runge-Kutta Framework and the Extension to Adaptivity in Time, J. Koellermeier, G. Samaey, in preparation
4) published: Shallow Water Moment models for bedload transport problems, J. Garres-Díaz, M. J. Castro, J. Koellermeier, T. Morales de Luna, Commun. in Comput. Phys., 30(3), 903-941, 2021;
Equilibrium Stability Analysis of Hyperbolic Shallow Water Moment equations, Q. Huang, J. Koellermeier, W.-A. Yong, Math. Method. Appl. Sci., 2022;
Recent developments in modeling free-surface flows with vertically-resolved velocity profiles using moments, J. Koellermeier, Proceedings of the 26th Congress of Differential Equations and Applications (CEDYA) and 16th Congress of Applied Mathematics (CMA) 2021, Institutional Repository of the Oviedo University, p.247-252 2021;
submitted: Steady States and Well-balanced Schemes for Shallow Water Moment Equations with Topography, J. Koellermeier, E. Pimentel-Garcia, submitted
The application of projective integration schemes for moment models has not been considered before. We showed that projective integration schemes speed up the computation of moment models by several orders of magnitude in applications like rarefied gases and free-surface flows. We expect further benefits from using our new spatially and space-time adaptive versions of the projective integration scheme for other applications. Revealing the relation of projective integration schemes and Runge-Kutta schemes opens many possibilities for the development of new schemes and we expect further progress on this topic in the future.
Furthermore, the derivation of new moment models for free-surface flows has gained attention of the research community. Tailoring these models to applications in river and coastal engineering is expected to deliver benefits in the future.
We hope that our results of this project will contribute to the application of moment models in computational science and engineering to gain insights in complex physical processes.
Furthermore, the derivation of new moment models for free-surface flows has gained attention of the research community. Tailoring these models to applications in river and coastal engineering is expected to deliver benefits in the future.
We hope that our results of this project will contribute to the application of moment models in computational science and engineering to gain insights in complex physical processes.