939 research outputs found

    Positivity-preserving methods for ordinary differential equations

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    [EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).Blanes Zamora, S.; Iserles, A.; Macnamara, S. (2022). Positivity-preserving methods for ordinary differential equations. ESAIM Mathematical Modelling and Numerical Analysis. 56(6):1843-1870. https://doi.org/10.1051/m2an/20220421843187056

    Lyapunov Stability of First and Second Order GeCo and gBBKS Schemes

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    In this paper we investigate the stability properties of fixed points of the so-called gBBKS and GeCo methods, which belong to the class of non-standard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. The schemes are applied to general linear test equations and proven to be generated by C1\mathcal C^1-maps with locally Lipschitz continuous first derivatives. As a result, a recently developed 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.Comment: 31 pages, 7 figure

    A discontinuous Galerkin method for the Vlasov-Poisson system

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    A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86 figure

    Effiziente numerische Methoden zur Lösung von reaktiven Euler-Gleichungen für mehrere Spezies

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    This cumulative thesis is devoted to the efficient simulation of compressible chemically reactive flows with multiple species and reactions being involved. In addition, the mass-fraction based reactive Euler equations with multiple species can be used to describe two-phase flows with multiple 'components' (corresponding to 'species') in a diffuse-interface manner, with suitable equations of state or thermodynamical models being employed. Three numerical methods towards computational high-efficiency solution of the above equation system are proposed.Diese kumulative Doktorarbeit widmet sich der effizienten Simulation kompressibler chemisch reaktiver Strömungen, wo mehrere Arten und Reaktionen beteiligt sind. Darüber hinaus können die auf Massenfraktionen basierenden reaktiven Euler-Gleichungen für mehrere Spezies mit geeigneten Zustandsgleichungen oder thermodynamischen Modellen verwendet werden, um zweiphasige Strömungen mit mehreren "Komponenten" (entsprechend "Spezies") auf diffuse Weise zu beschreiben. Drei numerische Methoden zur numerischen hocheffizienten Lösung des obigen Gleichungssystems warden vorgeschlagen
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