Diagonally implicit general linear methods for ordinary differential equations

Butcher, John; Jackiewicz, Zdzisław

doi:10.1007/bf01990528

Cited by 72 publications

(42 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further study into GLMs is partly motivated by trying to answer the question as to whether GLMs can more effectively combine algebraic stability with high stage order and efficient stage calculations. When 'algebraic stability' is replaced by the weaker requirement of 'A-stability,' this question has been answered positively by the work of Butcher and Jackiewicz [6,7,10]. However, as shown in [14], A-stability may not be a sufficiently strong stability condition in some circumstances.…”

Section: Introductionmentioning

confidence: 94%

“…In general, the order conditions for general linear methods are somewhat complicated, even when studied in isolation, see Butcher [4]. However, the order conditions simplify considerably when stage order q is close to the order p. Theorem 6.1 [6,9] Necessary and sufficient conditions for order p and stage order q, when p ≤ q + 1, are the existence of a vector function, φ : C → C r , such that Suppose now that the hypotheses of Theorem 6.1 are satisfied, and that φ can be expressed in the form…”

Section: Introductionmentioning

confidence: 98%

“…The second objective is made more feasible by applying the simplified order conditions [6,7,9] available when p ≤ q + 1. These conditions allow more freedom than those arising from collocation, and make sense when constructing methods for stiff problems, where the effective order is close to q.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algebraically stable general linear methods and the G-matrix

Hewitt

Hill

2009

Bit Numer Math

View full text Add to dashboard Cite

The standard algebraic stability condition for general linear methods (GLMs) is considered in a modified form, and connected to a branch of Control Theory concerned with the discrete algebraic Riccati equation (DARE). The DARE theory shows that, for an algebraically stable method, there is a minimal G-matrix, G * , satisfying an equation, rather than an inequality. This result, and another alternative reformulation of algebraic stability, are applied to construct new GLMs with 2 steps and 2 stages, one of which has order p = 4 and stage order q = 3. The construction process is simplified by method-equivalence, and Butcher's simplified order conditions for the case p ≤ q + 1.

show abstract

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Algebraically stable general linear methods and the G-matrix

Hewitt

Hill

2009

Bit Numer Math

View full text Add to dashboard Cite

show abstract

“…The methods we consider will be diagonally implicit in the sense that A has this structure. Note that alternatives to the special case in which p, q, r and s are equal, are discussed in [3].…”

Section: Y'(x) = F(x Y(x))mentioning

confidence: 98%

A transformation for the analysis of DIMSIMs

Butcher

1994

BIT

Self Cite

View full text Add to dashboard Cite

Abstract.The special case of diagonally-implicit multistage integration methods is considered in which the order, the stage order, the number of values passed between steps and the number of stages in a step, all coincide. It is shown that a similarity transformation can be applied to the matrices characterizing the method so as to simplify the expression for the stability polynomial and thus aid in the search for methods with acceptable stability, AMS Subject Classification: 65L05.

show abstract

“…Many approaches for parallel integration methods use parallel iteration schemes for non-parallel methods, e.g., [2,6,1]. A recent survey on classes of General Linear Methods with inherent parallelism can be found in [3].…”

Section: Introductionmentioning

confidence: 99%

Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

Schmitt

Weiner²,

Podhaisky

2005

Bit Numer Math

View full text Add to dashboard Cite

Peer two-step W-methods are designed for integration of stiff initial value problems with parallelism across the method. The essential feature is that in each time step s 'peer' approximations are employed having similar properties. In fact, no primary solution variable is distinguished. Parallel implementation of these stages is easy since information from one previous time step is used only and the different linear systems may be solved simultaneously. This paper introduces a subclass having order s−1 where optimal damping for stiff problems is obtained by using different system parameters in different stages. Favourable properties of this subclass are uniform stability for realistic stepsize sequences and a superconvergence property which is proved using a polynomial collocation formulation. Numerical tests on a shared memory computer of a matrix-free implementation with Krylov methods are included. (2000): 65L06, 65Y05. AMS subject classification

show abstract

Diagonally implicit general linear methods for ordinary differential equations

Cited by 72 publications

References 6 publications

Algebraically stable general linear methods and the G-matrix

Algebraically stable general linear methods and the G-matrix

A transformation for the analysis of DIMSIMs

Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration

Contact Info

Product

Resources

About