On implicit Runge-Kutta methods with high stage order

1997 ◽  
Vol 37 (1) ◽  
pp. 221-226 ◽  
Author(s):  
C. Bendtsen
Keyword(s):  
High Stage ◽  
Stage Order ◽  
Runge Kutta

scholarly journals On order 5 symplectic explicit Runge-Kutta Nyström methods

2000 ◽  
Vol 4 (2) ◽  
pp. 143-150 ◽  
Author(s):  
Lin-Yi Chou ◽  
P. W. Sharp
Keyword(s):  
Extra Cost ◽  
Optimal Method ◽  
New Methods ◽  
Nyström Methods ◽  
Stage Order ◽  
Principal Error ◽  
Numerical Performance ◽  
Free Parameters ◽  
Runge Kutta ◽  
Error Coefficients

Order five symplectic explicit Runge-Kutta Nyström methods of five stages are known to exist. However, these methods do not have free parameters with which to minimise the principal error coefficients. By adding one derivative evaluation per step, to give either a six-stage non-FSAL family or a seven-stage FSAL family of methods, two free parameters become available for the minimisation. This raises the possibility of improving the efficiency of order five methods despite the extra cost of taking a step.We perform a minimisation of the two families to obtain an optimal method and then compare its numerical performance with published methods of orders four to seven. These comparisons along with those based on the principal error coefficients show the new method is significantly more efficient than the five-stage, order five methods. The numerical comparisons also suggest the new methods can be more efficient than published methods of other orders.

scholarly journals Rational functions with maximal radius of absolute monotonicity

2014 ◽  
Vol 17 (1) ◽  
pp. 159-205 ◽  
Author(s):  
Lajos Lóczi ◽  
David I. Ketcheson
Keyword(s):  
Initial Value Problems ◽  
Rational Functions ◽  
Strong Stability ◽  
Strong Stability Preserving ◽  
Algebraic Numbers ◽  
Initial Value ◽  
Stage Order ◽  
Absolute Monotonicity ◽  
Runge Kutta ◽  
Radius Of Absolute Monotonicity

AbstractWe study the radius of absolute monotonicity $R$ of rational functions with numerator and denominator of degree $s$ that approximate the exponential function to order $p$. Such functions arise in the application of implicit $s$-stage, order $p$ Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$ and $R>2s$, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.

Explicit Runge–Kutta pairs with lower stage-order

2013 ◽  
Vol 65 (3) ◽  
pp. 555-577 ◽  
Author(s):  
J. H. Verner
Keyword(s):  
Lower Stage ◽  
Stage Order ◽  
Runge Kutta

scholarly journals Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Haiyan Yuan ◽  
Cheng Song
Keyword(s):  
Differential Equations ◽  
Quadrature Formula ◽  
Nonlinear Stability ◽  
Stability And Convergence ◽  
Asymptotically Stable ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Stage Order ◽  
Runge Kutta ◽  
The Stability

This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of(k,l)-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a(k,l)-algebraically stable two-step Runge-Kutta method with0<k<1is proved. For the convergence, the concepts ofD-convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order isp, then the method with compound quadrature formula isD-convergent of order at leastmin{p,ν}, whereνdepends on the compound quadrature formula.

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