2,064 research outputs found

    Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems

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    Explicit Runge-Kutta schemes with large stable step sizes are developed for integration of high order spectral difference spatial discretization on quadrilateral grids. The new schemes permit an effective time step that is substantially larger than the maximum admissible time step of standard explicit Runge-Kutta schemes available in literature. Furthermore, they have a small principal error norm and admit a low-storage implementation. The advantages of the new schemes are demonstrated through application to the Euler equations and the linearized Euler equations.Comment: 37 pages, 3 pages of appendi

    Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order

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    When evolving in time the solution of a hyperbolic partial differential equation, it is often desirable to use high order strong stability preserving (SSP) time discretizations. These time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. While the allowable time-step depends on both the spatial and temporal discretizations, the contribution of the temporal discretization can be isolated by taking the ratio of the allowable time-step of the high order method to the forward Euler time-step. This ratio is called the strong stability coefficient. The search for high order strong stability time-stepping methods with high order and large allowable time-step had been an active area of research. It is known that implicit SSP Runge-Kutta methods exist only up to sixth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge-Kutta methods of any linear order. In the current work we aim to find very high linear order implicit SSP Runge-Kutta methods that are optimal in terms of allowable time-step. Next, we formulate an optimization problem for implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge-Kutta methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs that have high linear order and nonlinear orders p=2,3,4. These methods are then tested on sample problems to verify order of convergence and to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems

    Optimal stability polynomials for numerical integration of initial value problems

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    We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied

    Optimal Explicit Strong Stability Preserving Runge--Kutta Methods with High Linear Order and optimal Nonlinear Order

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    High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge--Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge--Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge--Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may be still be worthwhile to use methods with higher nonlinear order
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