Numerical Methods for Compressible and Incompressible Flows 


I am working on numerical methods for compressible and incompressible flows. I focus on Finite Difference (FD) and Finite-Element Based Schemes (FE) like Continuous and Discontinous Galerkin (CG/DG), Flux Reconstruction (FR) and Residual Distribution (RD). I analyze the stability and approximation properties of these schemes. One major tool 

are summation-by-parts operators to transfrom results from the continuous setting to the discrete framework. Further, I also develop new stable methods for advection-dominated problems.

Research Highlights:

  • Development of new stable DG schemes and FR/RD schemes on polygonial meshes 
  • Investigation and reinterpretation of entropy correction terms and the application to fluid flows 
  • Study and construction of new boundary operators for FE based schemes  
  • Analyizing the long-term error behaviour of  DG and FR

Time Integration Schemes  


Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability without a time integration method is studied. In my reserach, I focus on this task and construct fully discrete entropy stable schemes. By applying the relaxation technique proposed by D. Ketcheson or the application of modal filters/artificial viscosity, I am able to construct fully discrete, explicit entropy stable schemes. Besides this, I am investigating stability properties of DeC, Ader and RK methods and focus on the relations between these schemes. 

As another point, I am working on positivity preserving time integration methods and was able to develop an

arbitrary high-order, conservative and positivity preserving modified Patankar DeC scheme. In the future, I will investigate the stability properties of these modified Patankar schemes. 

Research Highlights:

  • Stability investigation and reinterpretation of DeC, ADER and RK 
  • Development of arbitrary high order, positivity preserving and conservative modified Patankar DeC Schemes
  • Stability analysis for modified Patanker schemes
  • Construction of fully discrete (entropy) stable schemes 

Dissipative Solutions and K-Convergence 


The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy) weak solutions. Therefore, the concept of measure valued solutions seems nowadays more suitable for the analysis. In Mainz, I study the convergence of entropy stable FE based schemes for the complete compressible Euler equations in the multidimensional case and try to show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Therefore, I have to work on stability and consistency estimates in this context. 

Research Highlights: 

  • Convergence results for DG/RD schemes (hopefully)

Uncertainty Quantification 

I have already worked on UQ in the context of hyperbolic problems. I applied the polynomial chaos approach for the Burgers’ Equation and considered SBP-FR/DG methods in this context. We were able to construct entropy stable numerical fluxes for the first time (up to my knowledge). In the future, I will concentrate more on this topic and work on real applications. Since the resulting systems will be huge and shocks appear, shock sensors and model order reduction techniques will be part of my future research.

Research Highlights:

  • Construction of entropy stable numerical fluxes for FR methods using the polynomial chaos approach for Burgers' equation
  • Investigation of non-strictly hyperbolic systems 

Publications 



Published Articles (Journals): 


  1. J. Glaubitz, E. le Mèlèdo, P. Öffner - Towards Stable Radial Basis Function Methods for Linear Advection Problems - Computers and Mathematics with Applications, 2021 (accepted). (Preprint). 
  2. R. Abgrall, J. Nordström, P. Öffner, S. Tokareva - Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability - Communications on Applied Mathematics and Computation, 2021.  (doi).
  3. M. Han Veiga, P. Öffner, D.Torlo - DeC and ADER: Similarities, Differences, and a Unified Framework - Journal of Scientific Computing, 2020 (accepted). (arXiv).
  4. R. Abgrall, J. Nordström, P. Öffner, S. Tokareva - Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems - Journal of Scientific Computing 85, 43, 2020. (doi).
  5. E. le Mèlèdo, P. Öffner, R. Abgrall - General polytopial H(div) conformal finite elements and their discretisation spaces - ESAIM: Mathematical Modelling and Numerical Analysis, 2020 (accepted). (doi).

  6. P. Öffner, D. Torlo - Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes - Applied Numerical Mathematics 153, 15 - 34, 2020. (doi).

  7. 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).

  8. 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).

  9. P. Öffner, H. Ranocha - Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients - Journal of Scientific Computing 79(3), 1572 - 1607, 2019. (doi).

  10. 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).

  11. 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).

  12. J. Glaubitz, P. Öffner, Th. Sonar - Application of Modal Filtering to a Spectral Difference Method - Mathematics of Computation 87(309), 175–207, 2018. (doi).

  13. H. Ranocha, P. Öffner - L2 Stability of Explicit Runge-Kutta Schemes - Journal of Scientific Computing 75(2), 1040–1056, 2018. (doi).

  14. 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).

  15. H. Ranocha, P. Öffner, Th. Sonar - Summation-by-parts operators for correction procedure via reconstruction - Journal of Computational Physics 311, 299–328, 2016. (doi).

  16. P. Öffner, Th. Sonar, M. Wirz - Detecting strength and location of jump discontinuities in numerical data - Applied Mathematics 4, (12A), 1–14, 2013. (doi).

  17. P. Öffner, Th. Sonar - Spectral convergence for orthogonal polynomials on triangles - Numerische Mathematik 124 (4), 701–721, 2013. (doi).

Pre-prints and Technical Reports:


  1. R. Abgrall, E. Le Mèlèdo, P. Öffner, Davide Torlo - Relaxation Deferred Correction Methods  and their Applications to Residual Distribution Schemes - arXiv: 2106.05005, 2021. (arXiv).
  2. R. Abgrall, P. Öffner, H. Ranocha - Reinterpretation and Extension of Entropy Correction Terms for Residual Distribution and Discontinuous Galerkin Schemes - arXiv:1910.06783, 2020. (arXiv).

  3. R.Abgrall, E. le Mèlèdo, P. Öffner -A class of finite-dimensional spaces and H(div) conformal elements on general polytopes - arXiv:1907.08678, 2019. (arXiv).
  4. R. Abgrall, E. Le Mèlèdo, P. Öffner- On the Connection between Residual Distribution Schemes and Flux Reconstruction - hal-01820176, arXiv:1807.01261, 2019. (arXiv).
  5. P. Öffner - Error boundedness of Correction Procedure via Reconstruction / Flux Reconstruction - arXiv:1806.01575, 2019. (arXiv).
  6. R. Goertz, P. Öffner - On Hahn polynomial expansion of a continuous function of bounded variation - arXiv:1610.06748, 2016. (arXiv).

  7. R. Goertz, P. Öffner - Spectral accuracy for the Hahn polynomials - arXiv:1609.07291, 2016. (arXiv).


Published (Conference Proceedings):


  1. R. Abgrall, P. Öffner, H. Ranocha - Extension of Entropy Correction Terms for Residual Distribution Schemes: Application to Structure Preserving Discretization - Oberwolfach Report 11, 2021.
  2. M. Han Veiga, P. Öffner, D.Torlo - DeC and ADER: arbitrarily high order methods for hyperbolic PDEs (and ODEs) -Oberwolfach Report 11, 2021.
  3. 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.

  4. 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. 

  5. 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. 

  6. 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. 

  7. P. Öffner, Th. Sonar - Orthogonal Polynomials and their Application in a Spectral Difference Method - Oberwolfach Report 41, 2015. 

  8. 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. 



Theses: 


  • Approximation and Stability Properties of Numerical Methods for Hyperbolic Conservation Laws - Habilitation Thesis (submitted, 6.8.2020), University Zurich, 2020.

  • Two-dimensional classical and discrete orthogonal polynomials and their applications to spectral methods to solve hyperbolic conservations laws - Dissertation, TU Braunschweig, 2015.