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Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order. In a prior paper (Meyer, Balsara and Aslam 2012 [36]) we have also presented an extensive test-suite showing that the RKL2 method works robustly in the presence of shocks in an anisotropically conducting, magnetized plasma.
Univ. Next, we formulate a novel optimization problem for implicit–explicit (IMEX) SSP Runge–Kutta methods and find optimized IMEX SSP Runge–Kutta pairs that have high linear order plin ≤ 7 and nonlinear orders up to p = 4. Computational Mathematics Dept. We call this a convex monotonicity preserving property and show by examples that it is very useful in parabolic problems with variable diffusion coefficients. 272, Applied Numerical Mathematics, Vol. 77, Issue 2, Journal of Computational and Applied Mathematics, Vol. Kennedy, Christopher A.; Carpenter, Mark H. Higueras, Inmaculada; Happenhofer, Natalie; Koch, Othmar. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods. Kubatko, Ethan J.; Yeager, Benjamin A.; Ketcheson, David I. Ascher, Uri M.; Ruuth, Steven J.; Spiteri, Raymond J. Ketcheson, David I.; Macdonald, Colin B.; Gottlieb, Sigal. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Optimal first order explicit SSP Runge-Kutta methods consist simply of repeated forward Euler steps. of Mathematics; Sandia National Lab. Total-variation-diminishing implicit–explicit Runge–Kutta methods for the simulation of double-diffusive convection in astrophysics, Stepsize Restrictions for the Total-Variation-Diminishing Property in General Runge--Kutta Methods, Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations, Une m�thode multipas implicite-explicite pour l'approximation des �quations d'�volution paraboliques, On Strong Stability Preserving Time Discretization Methods, Additive Runge–Kutta schemes for convection–diffusion–reaction equations, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Optimized strong stability preserving IMEX Runge–Kutta methods, Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations, Representations of Runge--Kutta Methods and Strong Stability Preserving Methods, Strong Stability Preserving Two-step Runge–Kutta Methods, Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes, Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods, Runge–Kutta methods with minimum storage implementations, Total-Variation-Diminishing Time Discretizations, Optimal implicit strong stability preserving Runge–Kutta methods, Implicit-Explicit Methods for Time-Dependent Partial Differential Equations, Strong Stability for Additive Runge–Kutta Methods, Implicit-Explicit Runge--Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit, Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order, Characterizing Strong Stability Preserving Additive Runge-Kutta Methods, Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method, High-order, linearly stable, partitioned solvers for general multiphysics problems based on implicit–explicit Runge–Kutta schemes, Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. Similar Records in DOE PAGES and OSTI.GOV collections: This work introduces a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems from a monolithic implicit–explicit Runge–Kutta (IMEX-RK) discretization of the semi-discrete equations. x��\I���÷1?����R��
l�> iF[0�Z����˥�euv�X�A�JU�����-��1ˍ�ʟ��On��'�N��S�����l���7��Q�(1;������Yn���u'�����o�[�q��L[�g! First Order. Ascher, Uri M.; Ruuth, Steven J.; Wetton, Brian T. R. Gottlieb, Sigal; Grant, Zachary; Higgs, Daniel. 32, Issue 3, SIAM Journal on Numerical Analysis, Vol. treated by explicit Runge-Kutta methods and g is simultaneously treated by three implicit Runge-Kutta methods: a diagonally implicit Runge-Kutta method and two linearized implicit Runge-Kutta methods. (SNL-NM), Albuquerque, NM (United States). - Computer Methods in Applied Mechanics and Engineering. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s{sup 2} times larger than a, This paper is concerned with time-stepping numerical methods for computing stiff semi-discrete systems of ordinary differential equations for transient hypersonic flows with thermo-chemical non-equilibrium.
The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. This paper studies three different semi-implicit Runge-Kutta methods for additively split differential equations in the form of u{prime} = f(u) + g(u), where f is, Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order, Kupka, Friedrich; Happenhofer, Natalie; Higueras, Inmaculada. 5 0 obj and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. of New Mexico, Albuquerque, NM (United States). In the current work we find implicit SSP Runge–Kutta methods with high linear order plin ≤ 9 and nonlinear orders p = 2,3,4, that are optimal in terms of allowable SSP time-step. 9, Issue 6, Applied Numerical Mathematics, Vol. 68, Issue 3, Journal of Scientific Computing, Vol. ]��lwbV"z�����V�#z�No���Ye���jk�S��Az>`���N�fnzE����O��0�(�L*���ҳZ��m^��n�d>ίe]�������g>��v�ණs>�[ڊ�E�u+A�8�`��:w�);�����v�ӳW��)3#����ro+�h!���=��Tͨ������?�=/OH:�r��f�|��$}�j7��)�7ʆ�H�#����*��)L�AX�:�,S�0}A��E;}��I�97�io����S�#ب�A�������c�-d�D
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While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. The paper includes several details to facilitate implementation. To demonstrate the robustness and versatility of our methods, we show their successful operation on problems involving linear and non-linear heat conduction and viscosity, resistive magnetohydrodynamics, ambipolar diffusion dominated magnetohydrodynamics, level set methods and flux limited radiation diffusion. LINEAR CONSTANT COEFFICIENT SSP RUNGE–KUTTA METHODS OF ARBITRARY ORDER Although SSP methods were developed for use with nonlinear stability properties, they are equally useful in cases where the relevant operator is linear, and where linear norm properties are studied. In the current work we find implicit SSP Runge–Kutta methods with high linear order \(p_{lin} \le 9\) and nonlinear orders \(p=2,3,4\), that are optimal in terms of allowable SSP time-step. within the class of Runge–Kutta (RK) or linear multistep methods. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different, The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves.
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