Substantial extension and unification of the theory of Patankar-type schemes by means of unified order analysis, first-time investigation of stability, time-step adaptation and dense-output formulas.
Final Report Abstract
To describe real phenomena such as pandemic situations, chemical reactions or air and water flows, ordinary or partial differential equations are derived in mathematical modeling. As an analytical solution to these equations is usually not possible, numerical methods are used to approximate the solution. In view of the model assumptions and the errors in the measurement data, an exact representation of reality cannot be expected anyway. Instead, the aim of the numerical method is to retain as many physical properties of the underlying process as possible while providing approximations within the measurement accuracy. Important physical properties are, for example, the conservation of energy or mass as well as the positivity of certain solution components. For example, a numerical model for ocean currents without mass conservation would be misleading, as would an approximation of water height that takes negative values. These properties – conservativity and positivity – are not only physically relevant, but also essential for numerical calculation. Negative approximations for positive systems can lead to incorrect results or even to the termination of the procedure. In addition, failure to maintain conservativity could lead to incorrect equilibrium states. Patankar-type schemes build a family of numerical methods that are unconditionally positivitypreserving and conservative. At the time the project started, third order methods of this type already existed, but the order conditions were the result of complex and technical calculations. In addition, the stability analyses of these methods were previously based exclusively on numerical experiments. The aim of the project was to unify the order analysis, to develop a theoretical basis for the stability analysis of these methods and to increase their efficiency. A breakthrough in this project was the generalization of the theory of colored Butcher trees to derive conditions for Patankar-type methods of arbitrarily high order. In addition, the theory of dynamical systems was used to prove a theorem that extends the existing stability analysis and can also be applied to non-linear methods such as Patankar-type methods. In particular, the stability of these methods was analyzed for the first time. Further results of the project concern the efficiency of the methods. A new procedure for optimizing the time-step control for methods with an embedded method and good stability properties was presented and applied to modified Patankar-Runge-Kutta methods up to order three. Furthermore, these methods were equipped with a dense output formula that provides approximations of the same accuracy for arbitrary intermediate points in time with little additional computational effort.
Publications
-
Recent Developments in the Field of Modified Patankar‐Runge‐Kutta‐methods. PAMM, 21(1).
Izgin, Thomas; Kopecz, Stefan & Meister, Andreas
-
Lyapunov Stability of third order SSPMPRK schemes (code).
J. Huang, T. Izgin, S. Kopecz, A. Meister & C.-W. Shu
-
Modified Patankar: Oscillations and Lyapunov Stability (code).
Thomas Izgin, Philipp Öffner & Davide Torlo
-
On Lyapunov stability of positive and conservative time integrators and application to second order modified Patankar–Runge–Kutta schemes. ESAIM: Mathematical Modelling and Numerical Analysis, 56(3), 1053-1080.
Izgin, Thomas; Kopecz, Stefan & Meister, Andreas
-
On the Stability of Unconditionally Positive and Linear Invariants Preserving Time Integration Schemes. SIAM Journal on Numerical Analysis, 60(6), 3029-3051.
Izgin, Thomas; Kopecz, Stefan & Meister, Andreas
-
A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System. PAMM, 22(1).
Izgin, Thomas; Kopecz, Stefan & Meister, Andreas
-
A study of the local dynamics of modified Patankar DeC and higher order modified Patankar–RK methods. ESAIM: Mathematical Modelling and Numerical Analysis, 57(4), 2319-2348.
Izgin, Thomas & Öffner, Philipp
-
A study of the local dynamics of MPDeC and higher order MPRK methods (code).
Thomas Izgin & Philipp Öffner
-
On the dynamics of first and second order GeCo and gBBKS schemes. Applied Numerical Mathematics, 193, 43-66.
Izgin, Thomas; Kopecz, Stefan; Martiradonna, Angela & Meister, Andreas
-
On the stability of strong-stability-preserving modified Patankar–Runge–Kutta schemes. ESAIM: Mathematical Modelling and Numerical Analysis, 57(2), 1063-1086.
Huang, Juntao; Izgin, Thomas; Kopecz, Stefan; Meister, Andreas & Shu, Chi-Wang
-
Order conditions for NSARK methods (code).
Thomas Izgin, David I. Ketcheson & Andreas Meister
-
Using bayesian optimization to design time step size controllers with application to modified patankar–runge–kutta methods.
Thomas Izgin & Hendrik Ranocha
-
A Necessary Condition for Non-Oscillatory and Positivity Preserving Time-Integration Schemes. SEMA SIMAI Springer Series, 121-131. Springer Nature Switzerland.
Izgin, Thomas; Öffner, Philipp & Torlo, Davide
-
A Unifying Theory for Runge–Kutta-like Time Integrators: Convergence and Stability. PhD thesis, Kassel, Universität Kassel, Fachbereich Mathematik und Naturwissenschaften, Institut für Mathematik
Thomas Izgin
-
On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters. Numerical Algorithms, 96(3), 1221-1242.
Izgin, Thomas; Kopecz, Stefan; Meister, Andreas & Schilling, Amandine
-
A Boot-Strapping Technique to Design Unconditionally Positive Dense Output Formulae for Modified Patankar-Runge-Kutta Methods. Communications on Applied Mathematics and Computation.
Izgin, Thomas
-
Order conditions for Runge–Kutta-like methods with solution-dependent coefficients. Communications in Applied Mathematics and Computational Science, 20(1), 29-66.
Izgin, Thomas; Ketcheson, David I. & Meister, Andreas
