7 research outputs found
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
Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge--Kutta methods
We construct a family of embedded pairs for optimal strong stability
preserving explicit Runge-Kutta methods of order to be used
to obtain numerical solution of spatially discretized hyperbolic PDEs. In this
construction, the goals include non-defective methods, large region of absolute
stability, and optimal error measurement as defined in [5,19]. The new family
of embedded pairs offer the ability for strong stability preserving (SSP)
methods to adapt by varying the step-size based on the local error estimation
while maintaining their inherent nonlinear stability properties. Through
several numerical experiments, we assess the overall effectiveness in terms of
precision versus work while also taking into consideration accuracy and
stability.Comment: 22 pages, 49 figure
Implicit-explicit multirate infinitesimal GARK methods
This work focuses on the development of a new class of high-order accurate
methods for multirate time integration of systems of ordinary differential
equations. Unlike other recent work in this area, the proposed methods support
mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to
allowing this slow time scale flexibility, the proposed methods utilize a
so-called `infinitesimal' formulation for the fast time scale through
definition of a sequence of modified `fast' initial-value problems, that may be
solved using any viable algorithm. We name the proposed class as
implicit-explicit multirate infinitesimal generalized-structure additive
Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we
prove that they may be viewed as specific instances of generalized-structure
additive Runge--Kutta (GARK) methods, and derive a set of order conditions on
the IMEX-MRI-GARK coefficients to guarantee both third and fourth order
accuracy for the overall multirate method. Additionally, we provide three
specific IMEX-MRI-GARK methods, two of order three and one of order four. We
conclude with numerical simulations on two multirate test problems,
demonstrating the methods' predicted convergence rates and comparing their
efficiency against both legacy IMEX multirate schemes and recent third and
fourth order implicit MRI-GARK methods