2,715 research outputs found
Design and Optimization of Explicit Runge-Kutta Formulas
A model of the pretzel knot is described. Explicit Runge-Kutta methods have been studied for over a century and have applications in the sciences as well as mathematical software such as Matlab\u27s ode45 solver. We have taken a new look at fourth- and fifth-order Runge-Kutta methods by utilizing techniques based on Gröbner bases to design explicit fourth-order Runge-Kutta formulas with step doubling and a family of (4,5) formula pairs that minimize the higher-order truncation error. Gröbner bases, useful tools for eliminating variables, also helped to reveal patterns among the error terms. A Matlab program based on step doubling was then developed to compare the accuracy and efficiency of fourth-order Runge-Kutta formulas with that of ode45
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence
Time integration of Fourier pseudo-spectral DNS is usually performed using
the classical fourth-order accurate Runge--Kutta method, or other methods of
second or third order, with a fixed step size. We investigate the use of
higher-order Runge-Kutta pairs and automatic step size control based on local
error estimation. We find that the fifth-order accurate Runge--Kutta pair of
Bogacki \& Shampine gives much greater accuracy at a significantly reduced
computational cost. Specifically, we demonstrate speedups of 2x-10x for the
same accuracy. Numerical tests (including the Taylor-Green vortex,
Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the
reliability and efficiency of the method. We also show that adaptive time
stepping provides a significant computational advantage for some problems (like
the development of a Rayleigh-Taylor instability) without compromising
accuracy
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
An efficient Runge-Kutta (4,5) pair
AbstractA pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step
An Efficient Runge-Kutta (4,5) pair
A pair of explicit Runge-Kutta formulas of orders 4 and 5 is derived. It is significantly more efficient than the Fehlberg and Dormand-Prince pairs, and by standard measures it is of at least as high quality. There are two independent estimates of the local error. The local error of the interpolant is, to leading order, a problem-independent function of the local error at the end of the step
A family of parallel Runge-Kutta pairs
AbstractIncreasing availability of parallel computers has recently spurred a substantial amount of research concerned with designing explicit Runge-Kutta methods to be implemented on such computers. Here, we discuss a family of methods that require fewer processors than methods presently available do, still achieving a similar speed-up. In particular, (5,6) and (6,7) pairs are derived, that require a minimum number of function evaluations on two and three processors, respectively
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