54 research outputs found
Extrapolation-based implicit-explicit general linear methods
For many systems of differential equations modeling problems in science and
engineering, there are natural splittings of the right hand side into two
parts, one non-stiff or mildly stiff, and the other one stiff. For such systems
implicit-explicit (IMEX) integration combines an explicit scheme for the
non-stiff part with an implicit scheme for the stiff part.
In a recent series of papers two of the authors (Sandu and Zhang) have
developed IMEX GLMs, a family of implicit-explicit schemes based on general
linear methods. It has been shown that, due to their high stage order, IMEX
GLMs require no additional coupling order conditions, and are not marred by
order reduction.
This work develops a new extrapolation-based approach to construct practical
IMEX GLM pairs of high order. We look for methods with large absolute stability
region, assuming that the implicit part of the method is A- or L-stable. We
provide examples of IMEX GLMs with optimal stability properties. Their
application to a two dimensional test problem confirms the theoretical
findings
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
Two-step Runge-Kutta methods with quadratic stability functions
We describe the construction of implicit two-step Runge-Kutta methods with stability properties determined by quadratic stability functions. We will aim for methods which are A-stable and L-stable and such that the coefficients matrix has a one point spectrum. Examples of methods of order up to eight are provided
Explicit two-step Runge-Kutta methods
summary:The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order the minimal number of stages for explicit TSRK method of order is equal to the minimal number of stages for explicit Runge-Kutta method of order . Numerical results are presented which demonstrate that constant step size TSRK can be both effectively and efficiently used in an Euler equation solver. Furthermore, a comparison with a variable step size formulation shows that in these solvers the variable step size formulation offers no advantages compared to the constant step size implementation
A new strategy for choosing the Chebyshevâgegenbauer parameters in a reconstruction based on asymptotic analysis
The Gegenbauer reconstruction method, first proposed by Gottlieb et. al. in 1992, has been considered a useful technique for reâexpanding finite series polynomial approximations while simultaneously avoiding Gibbs artifacts. Since its introduction many studies have analyzed the method's strengths and weaknesses as well as suggesting several applications. However, until recently no attempts were made to optimize the reconstruction parameters, whose careful selection can make the difference between spectral accuracies and divergent error bounds.
In this paper we propose asymptotic analysis as a method for locating the optimal Gegenbauer reconstruction parameters. Such parameters are useful to applications of this reconstruction method that either seek to bound the number of Gegenbauer expansion coefficients or to control compression ratios. We then illustrate the effectiveness of our approach with the results from some numerical experiments.
First published online: 09 jun 201
Efficient General Linear Methods of High Order with Inherent Quadratic Stability
We search for general linear methods with s internal stages and r = s + 1 external stages of order p = s + 1 and stage order q = s. We require that stability function of these methods has only two non-zero roots. This is achieved by imposing the so-called inherent quadratic stability conditions. Examples of such general linear methods which are A- and L-stable up to the order p = 8 and stage order q = p - 1 are derived
Extrapolated ImplicitâExplicit RungeâKutta Methods
We investigate a new class of implicitâexplicit singly diagonally implicit RungeâKutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicitâexplicit general linear methods
Natural Volterra Runge-Kutta methods
A very general class of Runge-Kutta methods for Volterra integral equations of the second kind is analyzed. Order and stage order conditions are derived for methods of order p and stage order q = p up to the order four. We also investigate stability properties of these methods with respect to the basic and the convolution test equations. The systematic search for A- and V0-stable methods is described and examples of highly stable methods are presented up to the order p = 4 and stage order q = 4
Optimization-based search for nordsieck methods of high order with quadratic stability polynomials
We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7
Extrapolated ImplicitâExplicit RungeâKutta Methods
We investigate a new class of implicitâexplicit singly diagonally implicit RungeâKutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicitâexplicit general linear methods
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