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1. Positivity-preserving and energy-dissipative finite difference schemes for the Fokker–Planck and Keller–Segel equations

   https://doi.org/10.1093/imanum/drac014  

   Hu, Jingwei; Zhang, Xiangxiong (May 2022, IMA Journal of Numerical Analysis)

   Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions. 

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2. Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

   https://doi.org/10.1515/cmam-2021-0215  

   Cockburn, Bernardo; Du, Shukai; Sánchez, Manuel A. (May 2022, Computational Methods in Applied Mathematics)

   Abstract We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces.These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy.We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy.Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time.The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system.The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time.The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables. 

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3. An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

   https://doi.org/10.1002/num.22274  

   Jiang, Yi; Liu, Hailiang (May 2018, Numerical Methods for Partial Differential Equations)

   In this article, we introduce an invariant‐region‐preserving (IRP) limiter for thep‐system and the corresponding viscousp‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to thep‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscousp‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes. 

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4. Hyperbolic Problems: Theory, Numerics and Applications (2018): On structure-preserving high order methods for conservation laws

   Liu, Hailiang (January 2020, AIMS on Applied Mathematics)

   Bressan, A; Lewicka, M; Wang, D.; Zheng, Y.X. (Ed.)

   In this paper we review the algorithm development in high order methods for some conservation laws. The emphasis is on our recent contribution in the study of two model classes: Fokker-Planck-type equations and hyperbolic conservation law systems. For the former we will review free-energy-satisfying and positivity-preserving schemes. For the later we will review the general invariant-region-preserving (IRP) limiter, and its application to high order methods for multi-dimensional hyperbolic systems of conservation laws. 

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5. Semi-implicit hybrid discrete (H$$^T_N$$) approximation of thermal radiative transfer

   https://doi.org/https://link.springer.com/article/10.1007/s10915-021-01686-7  

   R.G. McClarren, J.A. Rossmanith (November 2021, Journal of scientific computing)

   The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions TRT are challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P$$_N$$ (spherical harmonics) or the S$$_N$$ (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H$$^T_N$$) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P$$_N$$ and S$$_N$$, and indeed reduces to each of these approximations in various limits: H$$^1_N$$ $$\equiv$$ P$$_N$$ and H$$^T_0$$ $$\equiv$$ S$$_T$$. We prove that H$$^T_N$$ results in a system of hyperbolic partial differential equations for all $$T\ge 1$$ and $$N\ge 0$$. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop a numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge-Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H$$^T_N$$ ansatz and the numerical discretizations. 

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