9 research outputs found
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
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
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
We develop a high-order kinetic scheme for entropy-based moment models of a
one-dimensional linear kinetic equation in slab geometry. High-order spatial
reconstructions are achieved using the weighted essentially non-oscillatory
(WENO) method, and for time integration we use multi-step Runge-Kutta methods
which are strong stability preserving and whose stages and steps can be written
as convex combinations of forward Euler steps. We show that the moment vectors
stay in the realizable set using these time integrators along with a maximum
principle-based kinetic-level limiter, which simultaneously dampens spurious
oscillations in the numerical solutions. We present numerical results both on a
manufactured solution, where we perform convergence tests showing our scheme
converges of the expected order up to the numerical noise from the numerical
optimization, as well as on two standard benchmark problems, where we show some
of the advantages of high-order solutions and the role of the key parameter in
the limiter
Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part
For many systems of differential equations modeling problems in science and
engineering, there are often natural splittings of the right hand side into two
parts, one of which is non-stiff or mildly stiff, and the other part is stiff.
Such systems can be efficiently treated by a class of implicit-explicit (IMEX)
diagonally implicit multistage integration methods (DIMSIMs), where the stiff
part is integrated by implicit formula, and the non-stiff part is integrated by
an explicit formula. We will construct methods where the explicit part has
strong stability preserving (SSP) property, and the implicit part of the method
is -, or -stable. We will also investigate stability of these methods
when the implicit and explicit parts interact with each other. To be more
precise, we will monitor the size of the region of absolute stability of the
IMEX scheme, assuming that the implicit part of the method is -, or
-stable. Finally we furnish examples of SSP IMEX DIMSIMs up to the order
four with good stability properties
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