3,773 research outputs found
Optimal Explicit Strong Stability Preserving Runge--Kutta Methods with High Linear Order and optimal Nonlinear Order
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
Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order
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
A high order accurate bound-preserving compact finite difference scheme for scalar convection diffusion equations
We show that the classical fourth order accurate compact finite difference
scheme with high order strong stability preserving time discretizations for
convection diffusion problems satisfies a weak monotonicity property, which
implies that a simple limiter can enforce the bound-preserving property without
losing conservation and high order accuracy. Higher order accurate compact
finite difference schemes satisfying the weak monotonicity will also be
discussed
Strong Stability Preserving Two-Step Runge-Kutta Methods
We investigate the strong stability preserving (SSP) property of two-step Runge– Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple\ud
subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge–Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations
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