My current research centers around numerical methods primarily applied to compressible flows. I specifically concentrate on the implementation and analysis of finite difference (FD) and finite element schemes (FE) such as continuous and discontinuous Galerkin (CG/DG), flux reconstruction (FR), and residual distribution (RD) schemes.
My primary focus lies in examining the stability and approximation properties of these schemes. To aid in this analysis, I employ summation-by-parts (SBP) operators, which enable the translation of results from the continuous setting to the discrete framework. In this regard, we have expanded the existing theory by incorporating general function spaces, resulting in what we refer to as FSBP operators.
Using these FSBP operators, we have successfully developed energy-stable RBF methods tailored to hyperbolic problems. This advancement has allowed us to enhance the accuracy and stability of our numerical approaches in the context of compressible flow simulations. In the future,
we plan to investigate error bounds of the new approach and build adaptive and stable FSBP schemes.
The issue of stability holds significant importance when it comes to numerical methods applied to hyperbolic conservation laws and has garnered considerable attention. However, the existing studies often assume continuity in time, focusing solely on semidiscrete stability without considering the time integration method. In my research, I address this gap by emphasizing the construction of fully discrete entropy stable schemes.
To achieve this, I explore various techniques such as the relaxation technique or the implementation of modal filters/artificial viscosity. These approaches enable me to construct explicit entropy stable schemes that are fully discrete. Additionally, I delve into investigating the stability properties of DeC, Ader, and RK methods, with a specific emphasis on establishing connections between these schemes.
Another aspect of my work involves developing time integration methods that preserve positivity. I have successfully designed a modified Patankar DeC scheme that is conservative, preserves positivity, and can achieve arbitrary high-order accuracy.
Moving forward, my research will be dedicated to exploring the stability properties of these modified Patankar schemes, further enhancing our understanding of their behavior.
The Cauchy problem associated with the complete Euler system typically exhibits ill-posedness when considering admissible (entropy) weak solutions. Consequently, the utilization of measure-valued solutions has emerged as a more suitable approach for analysis in recent times. In my research at Mainz, I focus on investigating the existence of dissipative solutions and the convergence of entropy stable finite element (FE) based schemes applied to the complete compressible Euler equations in multidimensional scenarios.
My objective is to demonstrate that the Young measure generated by numerical solutions effectively represents a dissipative (measure-valued) weak solution of the Euler system. To achieve this, I must delve into stability and consistency estimations within this context. Moreover, I am expanding my investigation to encompass more intricate systems and explore alternative methods. By doing so, I aim to enhance our understanding of these complex systems and advance the current knowledge in this field.
M. Ciallella, L. Micalizzi, P. Öffner, and D. Torlo - Modified Patankar Deferred Correction WENO Code for Shallow Water Equations, 2021, (git).
D. Torlo, P. Öffner and H. Ranocha - Stability of Positivity Preserving Patankar-Type Schemes, 2021, (git).
R. Abgrall, P. Bacigaluppi, L. Micalizzi, P. Öffner, S. Tokareva, D. Torlo and F. Mojarrad -
Residual Distribution high order code, 2021, (git).
M. Han Veiga, P. Öffner, and D. Torlo - DeC is ADER, 2020, (git).
P. Öffner and D. Torlo - Deferred Correction Patankar scheme, 2019, (git).