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.
Research Highlights:
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.
Research Highlights:
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.
Research Highlights:
Uncertainty Quantification
I have previously conducted research on uncertainty quantification (UQ) within the realm of hyperbolic problems. Specifically, I employed the polynomial chaos approach to analyze Burgers' Equation, utilizing SBP-FR/DG methods. Remarkably, we achieved a groundbreaking feat by successfully constructing entropy stable numerical fluxes. Currently, my focus lies on expanding this work to encompass the Shallow Water (SW) system, aiming to develop similar advancements in this domain.
In the future, my research efforts will be primarily concentrated on this topic, with a particular emphasis on real-world applications. As the resulting systems are expected to be vast and characterized by the presence of shocks, my investigations will incorporate shock sensors and model order reduction techniques. These strategies will play a crucial role in managing the computational challenges posed by these large-scale systems and enhancing the efficiency of the numerical methods employed.
Research Highlights:
The list of authors of mathematical publications is mainly in alphabetical order.
P. Öffner, D. Torlo - Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes - Applied Numerical Mathematics 153, 15 - 34, 2020. (doi).
J. Glaubitz, P. Öffner - Stable discretisations of high order discontinuous Galerkin methods on equidistant and scattered points - Applied Numerical Mathematics 151, 98 - 118, 2020. (doi).
P. Öffner, J. Glaubitz, H. Ranocha - Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods - International Journal of Numerical Analysis and Modeling 17.3, 332 - 349, 2020. (doi).
P. Öffner, H. Ranocha - Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients - Journal of Scientific Computing 79(3), 1572 - 1607, 2019. (doi).
P. Öffner, J. Glaubitz, H. Ranocha - Polynomial Chaos Method for the Burgers’ Equation using Correction Procedure via Reconstruction with Summation-by-Parts Operators - ESAIM: Mathematical Modelling and Numerical Analysis 52(6), 2215 2245, 2018. (doi).
H. Ranocha, J. Glaubitz, P. Öffner, Th. Sonar - Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators - Applied Numerical Mathematics 128, 1–23, 2018. (doi).
J. Glaubitz, P. Öffner, Th. Sonar - Application of Modal Filtering to a Spectral Difference Method - Mathematics of Computation 87(309), 175–207, 2018. (doi).
H. Ranocha, P. Öffner - L2 Stability of Explicit Runge-Kutta Schemes - Journal of Scientific Computing 75(2), 1040–1056, 2018. (doi).
H. Ranocha, P. Öffner, Th. Sonar - Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians - Journal of Computational Physics 342, 13–28, 2017. (doi).
H. Ranocha, P. Öffner, Th. Sonar - Summation-by-parts operators for correction procedure via reconstruction - Journal of Computational Physics 311, 299–328, 2016. (doi).
P. Öffner, Th. Sonar, M. Wirz - Detecting strength and location of jump discontinuities in numerical data - Applied Mathematics 4, (12A), 1–14, 2013. (doi).
P. Öffner, Th. Sonar - Spectral convergence for orthogonal polynomials on triangles - Numerische Mathematik 124 (4), 701–721, 2013. (doi).
S.-Ch. Klein and P. Öffner - Entropy conservative high-order fluxes in the presence of boundaries - arXiv:2211.01171, 2022.(arXiv).
R. Abgrall, E. le Mèlèdo, P. Öffner, H. Ranocha - Error boundedness of Correction Procedure via Reconstruction / Flux Reconstruction and the Connection to Residual Distribution Schemes - Hyperbolic Problems: Theory, Numerics, Applications - Proceedings of HYP2018, 2020.
J. Glaubitz, P. Öffner, H. Ranocha, Th. Sonar - Artificial Viscosity for CPR Methods Using SBP Operators - Springer Proceedings in Mathematics and Statistics: Proceeding of the XVI International Conference on Hyperbolic Problems Theory, Numerics, Applications, Aachen, 363-375, 2016.
P. Öffner, H. Ranocha, Th. Sonar - Correction Procedure via Reconstruction Using Summation-by- Parts Operators - Springer Proceedings in Mathematics and Statistics: Proceeding of the XVI International Conference on Hyperbolic Problems Theory, Numerics, Applications, Aachen, 491-501, 2016.
H.Ranocha, P. Öffner, Th. Sonar - Summation-by-Parts and Correction Procedure via Reconstruction - Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Ed. by M. L. Bittencourt, N. A. Dumont, J. S. Hesthaven. Vol. 119. Lecture Notes in Computational Science and Engineering. Cham: Springer, 627-637, 2017.
P. Öffner, Th. Sonar - Orthogonal Polynomials and their Application in a Spectral Difference Method - Oberwolfach Report 41, 2015.
P. Öffner, Th. Sonar - Spectral Approximation with Appell Polynomials - NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: Proceeding of the International Conference on Numerical Analysis and Applied Mathematics, Halkidiki, 2011.
Approximation and Stability Properties of Numerical Methods for Hyperbolic Conservation Laws - Habilitation Thesis (submitted, 6.8.2020), University Zurich, Springer Spektrum, 2023. (Book).
Two-dimensional classical and discrete orthogonal polynomials and their applications to spectral methods to solve hyperbolic conservations laws - Dissertation, TU Braunschweig, 2015.
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).
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