5,918 research outputs found
High order semi-implicit multistep methods for time dependent partial differential equations
We consider the construction of semi-implicit linear multistep methods which
can be applied to time dependent PDEs where the separation of scales in
additive form, typically used in implicit-explicit (IMEX) methods, is not
possible. As shown in Boscarino, Filbet and Russo (2016) for Runge-Kutta
methods, these semi-implicit techniques give a great flexibility, and allows,
in many cases, the construction of simple linearly implicit schemes with no
need of iterative solvers. In this work we develop a general setting for the
construction of high order semi-implicit linear multistep methods and analyze
their stability properties for a prototype linear advection-diffusion equation
and in the setting of strong stability preserving (SSP) methods. Our findings
are demonstrated on several examples, including nonlinear reaction-diffusion
and convection-diffusion problems
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
Effective order strong stability preserving Runge–Kutta methods
We apply the concept of effective order to strong stability preserving (SSP) explicit Runge–Kutta methods. Relative to classical Runge–Kutta methods, effective order methods are designed to satisfy a relaxed set of order conditions, but yield higher order accuracy when composed with special starting and stopping methods. The relaxed order conditions allow for greater freedom in the design of effective order methods. We show that this allows the construction of four-stage SSP methods with effective order four (such methods cannot have classical order four). However, we also prove that effective order five methods—like classical order five methods—require the use of non-positive weights and so cannot be SSP. By numerical optimization, we construct explicit SSP Runge–Kutta methods up to effective order four and establish the optimality of many of them. Numerical experiments demonstrate the validity of these methods in practice
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
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