arXiv subjects

Thomas Izgin

Publications and source records attributed to Thomas Izgin.

16 recordsLinked to original sources

arXiv2026

Using Machine Learning to Enhance Hyperparameter Optimization in Pandemic Modeling: Case study of COVID-19 Dynamics in Ghana

In this study, five distinct COVID-19 models developed in different countries, each designed to reflect the prevailing epidemiological condition at the time of formulation, are examined. The models are reformulated while still maintaining their original structure, using their common transmissions from one compartment to the other. Modified Patankar-Runge-Kutta (MPRK) methods are then applied to approximate the solutions of the resulting system of nonlinear ordinary differential equations (ODEs) representing each model to produce unconditionally positive approximations and to preserve the conservative part of the ODEs. In particular, we incorporate the numerical solution into a cost function to improve the estimates for the non-autonomous model hyperparameters. In a first step we obtain piecewise constant parameters that fit real data. Later we perform a WENO reconstruction in a post-process to approximate the true time-dependent coefficients inside the ODEs. As a proof-of-concept, we apply our approach to improve the parameters of a paper concerned with modeling COVID-19 in Ghana, where we can make 5-day predictions within a 10% error range.

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arXiv2026

A Positivity-Preserving Relaxation Algorithm

We combine Patankar-type methods with suitable relaxation procedures that are capable of ensuring correct dissipation or conservation of functionals such as entropy or energy while producing unconditionally positive and conservative approximations. To that end, we adapt the relaxation algorithm to enforce positivity by using either ideas from the dense output framework when a linear invariant must be preserved, or simply a geometric mean if the only constraint is positivity preservation. The latter merely requires the solution of a scalar nonlinear equation while former results in a coupled linear-nonlinear system of equations. We present sufficient conditions for the solvability of the respective equations. Several applications in the context of ordinary and partial differential equations are presented, and the theoretical findings are validated numerically.

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arXiv2026

Flux-Balanced Patankar-type Schemes for the Compressible Euler Equations

Positivity preservation of key physical quantities in the context of fluid flows, such as density and internal energy, is an essential property of a numerical scheme as otherwise the solution lacks physical relevance and has a not well-defined equation of state. One time integration technique that is capable of preserving the positivity of quantities for every time step size is the Patankar-trick and its variants. However, in the context of the Euler equations of gas dynamics, we wonder whether the Patankar-trick should be applied to the density and total energy equations or only to one of them. In this work, we discuss one drawback of the schemes when blindly applied to every positive conserved variable and additionally point out how to overcome the issue by balancing the involved numerical fluxes correctly. To illustrate our findings, we investigate modified Patankar--Runge--Kutta (MPRK) schemes in the context of the compressible Euler equations with and without stiff source terms. We discover that it is beneficial to only apply the Patankar-trick in the density equation and to balance the remaining numerical fluxes consistently rather than applying the trick also to the energy equation. This leads also to the preservation of contact discontinuities. We perform numerical experiments to demonstrate that the accuracy of the methods is maintained while the performance of our approach is superior to the traditional application of MPRK schemes.

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arXiv2025

The Lax-Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws

For hyperbolic conservation laws, the famous Lax-Wendroff theorem delivers sufficient conditions for the limit of a convergent numerical method to be a weak (entropy) solution. This theorem is a fundamental result, and many investigations have been done to verify its validity for finite difference, finite volume, and finite element schemes, using either explicit or implicit linear time-integration methods. Recently, the use of modified Patankar (MP) schemes as time-integration methods for the discretization of hyperbolic conservation laws has gained increasing interest. These schemes are unconditionally conservative and positivity-preserving and only require the solution of a linear system. However, MP schemes are by construction nonlinear, which is why the theoretical investigation of these schemes is more involved. We prove an extension of the Lax-Wendroff theorem for the class of MP methods. This is the first extension of the Lax--Wendroff theorem to nonlinear time integration methods with just an additional hypothesis on the total time variation boundedness of the numerical solutions. We provide some numerical simulations that validate the theoretical observations.

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arXiv2024

A Boot-Strapping Technique to Design Dense Output Formulae for Modified Patankar-Runge-Kutta Methods

In this work modified Patankar-Runge-Kutta (MPRK) schemes up to order four are considered and equipped with a dense output formula of appropriate accuracy. Since these time integrators are conservative and positivity preserving for any time step size, we impose the same requirements on the corresponding dense output formula. In particular, we discover that there is an explicit first order formula. However, to develop a boot-strapping technique we propose to use implicit formulae which naturally fit into the framework of MPRK schemes. In particular, if lower order MPRK schemes are used to construct methods of higher order, the same can be donw with the dense output formulae we propose in this work. We explicitly construct formulae up to order three and demonstrate how to generalize this approach as long as the underlying Runge-Kutta method possesses a dense output formulae of appropriate accuracy. We also note that even though linear systems have to be solved to compute an approximation for intermediate points in time using these higher order dense output formulae, the overall computational effort is reduced compared to using the scheme with a smaller step size.

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arXiv2024

A Unifying Theory for Runge-Kutta-like Time Integrators: Convergence and Stability

The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. The first major part of this thesis is dedicated to providing a tool for deriving order conditions for NSARK methods. The proposed approach may yield implicit order conditions, which can be rewritten in explicit form using the NB-series of the stages. The obtained explicit order conditions can be further reduced using Gr\"obner bases computations. With the presented approach, it was possible for the first time to obtain conditions for the construction of 3rd and 4th order GeCo as well as 4th order MPRK schemes. Moreover, a new fourth order MPRK method is constructed using our theory and the order of convergence is validated numerically. The second major part is concerned with the stability of nonlinear time integrators preserving at least one linear invariant. We discuss how the given approach generalizes the notion of A-stability. We can prove that investigating the Jacobian of the generating map is sufficient to understand the stability of the nonlinear method in a neighborhood of the steady state. This approach allows for the first time the investigation of several modified Patankar. In the case of MPRK schemes, we compute a general stability function in a way that can be easily adapted to the case of PDRS. Finally, the approach from the theory of dynamical systems is used to derive a necessary condition for avoiding unrealistic oscillations of the numerical approximation.

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arXiv2023

Using Machine Learning to Design Time Step Size Controllers for Stable Time Integrators

We present a new method for developing time step controllers based on a technique from the field of machine learning. This method is applicable to stable time integrators that have an embedded scheme, i.e., that have local error estimation similar to Runge-Kutta pairs. To design good time step size controllers using these error estimates, we propose to use Bayesian optimization. In particular, we design a novel objective function that captures important properties such as tolerance convergence and computational stability. We apply our new approach to several modified Patankar--Runge--Kutta (MPRK) schemes and a Rosenbrock-type scheme, equipping them with controllers based on digital signal processing which extend classical PI and PID controllers. We demonstrate that the optimization process yields controllers that are at least as good as the best controllers chosen from a wide range of suggestions available for classical explicit and implicit time integration methods by providing work-precision diagrams on a variety of ordinary and partial differential equations.

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arXiv2023

On the non-global linear stability and spurious fixed points of MPRK schemes with negative RK parameters

Recently, a stability theory has been developed to study the linear stability of modified Patankar--Runge--Kutta (MPRK) schemes. This stability theory provides sufficient conditions for a fixed point of an MPRK scheme to be stable as well as for the convergence of an MPRK scheme towards the steady state of the corresponding initial value problem, whereas the main assumption is that the initial value is sufficiently close to the steady state. Initially, numerical experiments in several publications indicated that these linear stability properties are not only local, but even global, as is the case for general linear methods. Recently, however, it was discovered that the linear stability of the MPDeC(8) scheme is indeed only local in nature. Our conjecture is that this is a result of negative Runge--Kutta (RK) parameters of MPDeC(8) and that linear stability is indeed global, if the RK parameters are nonnegative. To support this conjecture, we examine the family of MPRK22($α$) methods with negative RK parameters and show that even among these methods there are methods for which the stability properties are only local. However, this local linear stability is not observed for MPRK22($α$) schemes with nonnegative Runge-Kutta parameters.

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arXiv2023

Order conditions for Runge--Kutta-like methods with solution-dependent coefficients

In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent factors. These include the classes of modified Patankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods. Compared to traditional RK methods, the analysis of accuracy and stability of these methods is more complicated. In this work, we provide a comprehensive and unifying theory of order conditions for such RK-like methods, which differ from original RK schemes in that their coefficients are solution-dependent. The resulting order conditions are themselves solution-dependent and obtained using the theory of NB-series, and thus, can easily be read off from labeled N-trees. We present for the first time order conditions for MPRK and GeCo schemes of arbitrary order; For MPRK schemes, the order conditions are given implicitly in terms of the stages. From these results, we recover as particular cases all known order conditions from the literature for first- and second-order GeCo as well as first-, second- and third-order MPRK methods. Additionally, we derive sufficient and necessary conditions in an explicit form for 3rd and 4th order GeCo schemes as well as 4th order MPRK methods. We also present a new 4th order MPRK method within this framework and numerically confirm its convergence rate.

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arXiv2023

A study of the local dynamics of modified Patankar DeC and higher order modified Patankar RK methods

Patankar schemes have attracted increasing interest in recent years because they preserve the positivity of the analytical solution of a production-destruction system (PDS) irrespective of the chosen time step size. Although they are now of great interest, for a long time it was not clear what stability properties such schemes have. Recently a new stability approach based on Lyapunov stability with an extension of the center manifold theorem has been proposed to study the stability properties of positivity-preserving time integrators. In this work, we study the stability properties of the classical modified Patankar--Runge--Kutta schemes (MPRK) and the modified Patankar Deferred Correction (MPDeC) approaches. We prove that most of the considered MPRK schemes are stable for any time step size and compute the stability function of MPDeC. We investigate its properties numerically revealing that also most MPDeC are stable irrespective of the chosen time step size. Finally, we verify our theoretical results with numerical simulations.

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arXiv2023

On the Dynamics of First and Second Order GeCo and gBBKS Schemes

In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes. For the analysis, the schemes are applied to general linear equations and proven to be generated by $\mathcal C^1$-maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge--Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.

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arXiv2022

A necessary condition for non oscillatory and positivity preserving time-integration schemes

Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration schemes constructed for production-destruction systems. For such schemes, a classical stability analysis does not yield any information about the performance. Recently, two different techniques have been proposed to investigate the properties of MP schemes. In Izgin et al. [ESAIM: M2AN, 56 (2022)], inspired from dynamical systems, the Lyapunov stability properties of such schemes have been investigated, while in Torlo et al. [Appl. Numer. Math., 182 (2022)] their oscillatory behaviour has been studied. In this work, we investigate the connection between the oscillatory behaviour and the Lyapunov stability and we prove that a condition on the Lyapunov stability function is necessary to avoid oscillations. We verify our theoretical result on several numerical tests.

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arXiv2022

A Stability Analysis of Modified Patankar-Runge-Kutta methods for a nonlinear Production-Destruction System

Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i.e. the iterates are generated by a nonlinear map $\mathbf g$ even when the PDS is linear. Moreover, due to the conservativity of the method, the map $\mathbf g$ possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.

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arXiv2022

On the stability of strong-stability-preserving modified Patankar Runge-Kutta schemes

In this paper, we perform stability analysis for a class of second and third order accurate strong-stability-preserving modified Patankar Runge-Kutta (SSPMPRK) schemes, which were introduced in [4,5] and can be used to solve convection equations with stiff source terms, such as reactive Euler equations, with guaranteed positivity under the standard CFL condition due to the convection terms only. The analysis allows us to identify the range of free parameters in these SSPMPRK schemes in order to ensure stability. Numerical experiments are provided to demonstrate the validity of the analysis.

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arXiv2022

On the Stability of Unconditionally Positive and Linear Invariants Preserving Time Integration Schemes

Higher-order time integration methods that unconditionally preserve the positivity and linear invariants of the underlying differential equation system cannot belong to the class of general linear methods. This poses a major challenge for the stability analysis of such methods since the new iterate depends nonlinearly on the current iterate. Moreover, for linear systems, the existence of linear invariants is always associated with zero eigenvalues, so that steady states of the continuous problem become non-hyperbolic fixed points of the numerical time integration scheme. Altogether, the stability analysis of such methods requires the investigation of non-hyperbolic fixed points for general nonlinear iterations. Based on the center manifold theory for maps we present a theorem for the analysis of the stability of non-hyperbolic fixed points of time integration schemes applied to problems whose steady states form a subspace. This theorem provides sufficient conditions for both the stability of the method and the local convergence of the iterates to the steady state of the underlying initial value problem. This theorem is then used to prove the unconditional stability of the MPRK22($α$)-family of modified Patankar-Runge-Kutta schemes when applied to arbitrary positive and conservative linear systems of differential equations. The theoretical results are confirmed by numerical experiments.

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arXiv2022

On Lyapunov Stability of Positive and Conservative Time Integrators and Application to Second Order Modified Patankar--Runge--Kutta Schemes

Since almost twenty years, modified Patankar--Runge--Kutta (MPRK) methods have proven to be efficient and robust numerical schemes that preserve positivity and conservativity of the production-destruction system irrespectively of the time step size chosen. Due to these advantageous properties they are used for a wide variety of applications. Nevertheless, until now, an analytic investigation of the stability of MPRK schemes is still missing, since the usual approach by means of Dahlquist's equation is not feasible. Therefore, we consider a positive and conservative 2D test problem and provide statements usable for a stability analysis of general positive and conservative time integrator schemes based on the center manifold theory. We use this approach to investigate the Lyapunov stability of the second order MPRK22($α$) and MPRK22ncs($α$) schemes. We prove that MPRK22($α$) schemes are unconditionally stable and derive the stability regions of MPRK22ncs($α$) schemes. Finally, numerical experiments are presented, which confirm the theoretical results.

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