Class Of General Linear Methods Research Articles

On the dynamics of first and second order GeCo and gBBKS schemes

In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes.For the analysis, the schemes are applied to general linear equations and proven to be generated by C1-maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it.In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge–Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.

A Stability Analysis of Modified Patankar–Runge–Kutta methods for a nonlinear Production–Destruction System

AbstractModified Patankar–Runge–Kutta (MPRK) methods preserve the positivity as well as conservativity of a production–destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i. e. the iterates are generated by a nonlinear map g even when the PDS is linear. Moreover, due to the conservativity of the method, the map g possesses non‐hyperbolic fixed points.Recently, a new theorem for the investigation of stability properties of non‐hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.

A GLMs-based difference-quadrature scheme for Volterra integro-differential equations

In this paper, a class of general linear methods is combined with a special quadrature rule inherited from natural Runge–Kutta methods for Volterra integral equations to design a more stable and efficient algorithm. The constructed methods of orders 1,2, and 3 are A0–stable and the method of order 4 is A0(α)–stable. The capability of the proposed methods in solving stiff and nonstiff problems is shown by some numerical experiments.

Towards a code for nonstiff differential systems based on general linear methods with inherent Runge–Kutta stability

Various issues are discussed which are relevant to the development of the code for nonstiff differential systems based on a class of general linear methods with inherent Runge–Kutta stability.

Error Inhibiting Block One-step Schemes for Ordinary Differential Equations

The commonly used one step methods and linear multi-step methods all have a global error that is of the same order as the local truncation error (as defined in [1, 6, 8, 13, 15]). In fact, this is true of the entire class of general linear methods. In practice, this means that the order of the method is typically defined solely by order conditions which are derived by studying the local truncation error. In this work we investigate the interplay between the local truncation error and the global error, and develop a methodology which defines the construction of explicit error inhibiting block one-step methods (alternatively written as explicit general linear methods [2]). These error inhibiting schemes are constructed so that the accumulation of the local truncation error over time is controlled, which results in a global error that is one order higher than the local truncation error. In this work, we delineate how to carefully choose the coefficient matrices so that the growth of the local truncation error is inhibited. We then use this theoretical understanding to construct several methods that have higher order global error than local truncation error, and demonstrate their enhanced order of accuracy on test cases. These methods demonstrate that the error inhibiting concept is realizable. Future work will further develop new error inhibiting methods and will analyze the computational efficiency and linear stability properties of these methods.

Order reduction phenomenon for general linear methods

The order reduction phenomenon for general linear methods (GLMs) for stiff differential equations is investigated. It turns out that, similarly as for standard Runge–Kutta methods, the effective order of convergence for a large class of GLMs applied to stiff differential systems, is equal to the stage order of the method. In particular, it is demonstrated that the global error ‖e[n]‖ of GLMs of order p and stage order q applied to the Prothero–Robinson test problem y′(t)=λ(y(t)−φ(t))+φ′(t), t∈[t0,T], y(t0)=φ(t0), is O(hq)+O(hp) as h→0 and hλ→−∞. Moreover, for GLMs with Runge–Kutta stability which are A(0)-stable and for which the stability function R(z) of the underlying Runge–Kutta methods, (i.e., the corresponding RK methods which have the same absolute stability properties as the GLMs), is such that R(∞)≠1, the global error satisfies ‖e[n]‖=O(hq+1)+O(hp) as h→0 and hλ→−∞. These results are confirmed by numerical experiments.

Strong stability preserving explicit peer methods

In this paper we study explicit peer methods up to order p=13 which have the strong stability preserving (SSP) property. This class of general linear methods has the favourable property of a high stage order. The effective SSP coefficient is maximized by solving a nonlinear constraint optimization problem numerically to high precision. The coefficient matrices of the optimized methods are sparse in a very structured way. Linear multistep methods are obtained as a special case of only one stage.

STRONG STABILITY PRESERVING MULTISTAGE INTEGRATION METHODS

In this paper we systematically investigate explicit strong stability preserving (SSP) multistage integration methods, a subclass of general linear methods (GLMs), of order p and stage order q ≤ p. Characterization of this class of SSP GLMs is given and examples of SSP methods of order p ≤ 4 and stage order q = 1, 2, . . . , p are provided. Numerical tests are reported which confirm that the constructed methods achieve the expected order of accuracy and preserve monotonicity.

Explicit Runge–Kutta pairs with lower stage-order

Explicit Runge–Kutta pairs of methods of successive orders of accuracy provide effective algorithms for approximating solutions to nonstiff initial value problems. For each explicit RK method of order of accuracy p, there is a minimum number s p of derivative evaluations required for each step propagating the numerical solution. For p ≤ 8, Butcher has established exact values of s p , and for p > 8, his work establishes lower bounds; otherwise, upper bounds are established by various published methods. Recently, Khashin has derived some new methods numerically, and shown that the known upper bound on s 9 for methods of order p = 9 can be reduced from 15 to 13. His results motivate this attempt to identify parametrically exact representations for coefficients of such methods. New pairs of methods of orders 5,6 and 6,7 are characterized in terms of several arbitrary parameters. This approach, modified from an earlier one, increases the known spectrum of types of RK pairs and their derivations, may lead to the derivation of new RK pairs of higher-order, and possibly to other types of explicit algorithms within the class of general linear methods.

Construction of algebraically stable DIMSIMs

The class of general linear methods for ordinary differential equations combines the advantages of linear multistep methods (high efficiency) and Runge–Kutta methods (good stability properties such as A-, L-, or algebraic stability), while at the same time avoiding the disadvantages of these methods (poor stability of linear multistep methods, high cost for Runge–Kutta methods). In this paper we describe the construction of algebraically stable general linear methods based on the criteria proposed recently by Hewitt and Hill. We also introduce the new concept of ϵ-algebraic stability and investigate its consequences. Examples of ϵ-algebraically stable methods are given up to order p=4.

Search for efficient general linear methods for ordinary differential equations

We analyze implicit general linear methods with s internal stages and r=s+1 external stages of order p=s+1 and stage order q=s or q=s+1. These methods might eventually lead to more efficient formulas than the class of DIMSIMs and the class of general linear methods with inherent Runge–Kutta stability. We analyze also error propagation and estimation of local discretization errors. Examples of such methods which are A- and L-stable are derived up to the stage order q=3 or q=4 and order p=4.

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q = p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.

Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems

For Runge–Kutta methods and linear multistep methods, much attention has been paid, in the literature, to special nonlinear stability properties indicated by the terms total-variation-diminishing (TVD), strong-stability-preserving (SSP), and monotonicity. Stepsize conditions, guaranteeing these properties, were studied, e.g., by Shu and Osher [J. Comput. Phys., 77 (1988), pp. 439–471], Gottlieb, Shu, and Tadmor [SIAM Rev., 43 (2001), pp. 89–112], Hundsdorfer and Ruuth [Monotonicity for Time Discretizations, Dundee Conference Report NA/217 2003, University of Dundee, Dundee, UK, 2003, pp. 85–94], Higueras [J. Sci. Comput., 21 (2004), pp. 193–223] and [SIAM J. Numer. Anal., 43 (2005), pp. 924–948], Spiteri and Ruuth [SIAM J. Numer. Anal., 40 (2002), pp. 469–491], Gottlieb [J. Sci. Comput., 25 (2005), pp. 105–128], and Ferracina and Spijker [SIAM J. Numer. Anal., 42 (2004), pp. 1073–1093] and [Math. Comp., 74 (2005), pp. 201–219]. In the present paper, we obtain a special stepsize condition guaranteeing the above properties, for a generic numerical process. This condition is best possible in a well defined and natural sense. It is applicable to the important class of general linear methods, and it can also be used to answer some open questions, for methods of which the above stability properties were studied earlier.

B-theory of general linear methods for Volterra functional differential equations

B-theory of general linear methods (GLMs) for nonlinear Volterra functional differential equations (VFDEs) is established, which provides unified theoretical foundation for the study of GLMs when applied to nonlinear initial value problems (IVPs) in stiff ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice, and can be regarded as extension of the B-theory of Runge–Kutta methods for VFDEs presented by the same author in a previous paper. The extension from Runge–Kutta methods to the much more general class of GLMs is of essential importance since there exist many B-stable methods which are not one-step Runge–Kutta methods but can be regarded as special cases of GLMs.

Nonlinear Stability of General Linear Methods for Delay Differential Equations

This paper is devoted to a study of nonlinear stability of general linear methods for the numerical solution of delay differential equations in Hilbert spaces. New stability concepts are further introduced. The stability properties of (k,p,q)-algebraically stable general linear methods with piecewise constant or linear interpolation procedure are investigated. We also discuss stability of linear multistep methods viewed as a special subset of the class of general linear methods.

Explicit General Linear Methods with Inherent Runge Kutta Stability

A class of general linear methods is derived for application to non-stiff ordinary differential equations. A property known as “inherent Runge–Kutta stability” guarantees the stability regions of these methods are the same as for Runge–Kutta methods. Methods with this property have high stage order which enables asymptotically correct error estimates and high order interpolants to be computed conveniently. Some preliminary numerical experiments are given comparing these methods with some well known Runge–Kutta methods.

Parallel Störmer–Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form y′′= f( y) . The most successful integration methods are based on high-order Runge–Kutta–Nyström formulas. However, these methods were designed for sequential computer systems. In this paper, we consider high-order parallel methods that are not based on Runge–Kutta–Nyström formulas, but which fit into the class of general linear methods. In each step, these methods compute blocks of k approximate solution values (or stage values) at k different points using the whole previous block of solution values. The k stage values can be computed in parallel, so that on a k-processor computer system such methods effectively perform as a one-value method. The block methods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Störmer–Cowell type. For k=4 and k=5 we constructed explicit PSC methods with stage order q= k and step point order p= k+1 and implicit PSC methods with q= k+1 and p= k+2. For k≥6 we can construct explicit PSC methods with q= k and p= k+2 and implicit PSC methods with q= k+1 and p= k+3. It turns out that for k≥5 the abscissae of the stage values can be chosen such that only k−1 stage values in each block have to be computed, so that the number of computational stages, and hence the number of processors and the number of starting values needed, reduces to k ∗=k−1 . The numerical examples reported in this paper show that the effective number of right-hand side evaluations required by a variable stepsize implementation of the 10th-order PSC method is 4 up to 30 times less than required by the Runge–Kutta–Nyström code DOPRIN (which is considered as one of the most efficient sequential codes for second-order ODEs). Furthermore, a comparison with the 12th-order parallel code PIRKN reveals that the PSC code is, in spite of its lower order, at least equally efficient, and in most cases more efficient than PIRKN.

DIMSEMs — Diagonally Implicit Single-Eigenvalue Methods for the Numerical Solution of Stiff ODEs on Parallel Computers

This paper derives a new class of general linear methods (GLMs) intended for the solution of stiff ordinary differential equations (ODEs) on parallel computers. Although GLMs were introduced by Butcher in the 1960s, the task of deriving formulas from the class with properties suitable for specific applications is far from complete. This paper is a contribution to that work. Our new methods have several properties suited for the solution of stiff ODEs on parallel computers. They are strictly diagonally implicit, allowing parallelism in the Newton iteration used to solve the nonlinear equations arising from the implicitness of the formula. The stability matrix has no spurious eigenvalues (that is, only one eigenvalue of the stability matrix is non-zero), resulting in a solution free from contamination from spurious solutions corresponding to non-dominant, non-zero eigenvalues. From these two properties arises the name DIMSEM, for Diagonally IMplicit Single-Eigenvalue Method. The methods have high stage order, avoiding the phenomenon of order reduction that occurs, for example, with some Runge-Kutta methods. The methods are L-stable, with the result that the chosen stepsize is dictated by convergence requirements rather than stability considerations imposed by the stiffness of the problem. An introduction to GLMs is given and some order barriers for DIMSEMs are presented. DIMSEMs of orders 2-6 are derived, as well as an L-stable class of diagonal methods of all orders which do not, however, possess the single-eigenvalue property. A fixed-order, variable-stepsize implementation of the DIMSEMs is described, including the derivation of local error estimators, and the results of testing on both sequential and parallel computers is presented. The testing shows the DIMSEMs to be competitive with fixed-order versions of the popular solver LSODE on a practical test problem.

General linear methods with external stages of different orders

The approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. In this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna.

Triangularly Implicit Iteration Methods for ODE-IVP Solvers

It often happens that iteration processes used for solving the implicit relations arising in ODE-IVP methods only start to converge rapidly after a certain number of iterations. Fast convergence right from the beginning is particularly important if we want to use so-called step-parallel iteration in which the iteration method is concurrently applied at a number of step points. In this paper, we construct highly parallel iteration methods that do converge fast from the first iteration on. Our starting point is the PDIRK method (parallel, diagonally implicit, iterated Runge--Kutta method), designed for solving implicit Runge--Kutta equations on parallel computers. The PDIRK method may be considered as a Newton-type iteration in which the Newton Jacobian is "simplified" to block-diagonal form. However, when applied in a step-parallel mode, it turns out that its relatively slow convergence, or even divergent behavior, reduces the effectiveness of the step-parallel scheme. By replacing the block-diagonal Newton Jacobian approximation in PDIRK by a block-triangular approximation, we do achieve convergence right from the beginning at a modest increase of the computational costs. Our convergence analysis of the block-triangular approach will be given for the wide class of general linear methods, but the derivation of iteration schemes is limited to Runge--Kutta-based methods. A number of experiments show that the new parallel, triangularly implicit, iterated Runge--Kutta method (PTIRK method) is a considerable improvement over the PDIRK method.