High Stage Order Research Articles

Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control

It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (3, 3) and (4, 3) in detail and provide three Peer triplets of practical interest. We prove convergence of order s-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s-1$$\\end{document}, at least, for s-stage methods if state, adjoint and control satisfy the corresponding order conditions. Numerical tests show the expected order of convergence for the new Peer triplets.

Avoiding order reduction phenomenon for general linear methods when integrating linear problems with time dependent boundary values

When applied to stiff problems, the effective order of convergence of general linear methods is governed by their stage order, which is less than or equal to the classical order of the method. This produces an order reduction phenomenon, present in all general linear methods except those with high stage order, in a manner similar to that observed in other time integrators with internal stages.In this paper, we investigate the order reduction which arises when general linear methods are used as time integrators when using the method of lines for solving numerically initial boundary value problems with time dependent boundary values.We propose a technique, based on making an appropriate choice of the boundary values for the internal stages, with which it is possible to recover one unit of order, as we prove in this work. As expected, this implies a considerable improvement for the general linear methods suffering order reduction. Moreover, numerical experiments show that the improvement is not only in these cases, but that, even when the order reduction is not expected, the size of the errors is drastically reduced by using the technique proposed in this paper.

Explicit two-step peer methods with reused stages

Two-step peer methods for the numerical solution of Initial Value Problems (IVP) combine the advantages of Runge-Kutta (RK) and multistep methods to obtain high stage order and provide in a natural way a dense output. In general, explicit s-stage peer methods require s evaluations of the vector field at each step. Nevertheless, Klinge and coworkers (BIT Numer Math, 2018) have shown that some methods use less function calls se<s, here called effective stages, by re-using sr=s−se previously computed stages (shifted stages) from the previous steps in the current one.In this paper we propose a new approach, different from the one used by Klinge and coworkers, to re-use previously computed stages, that we call peer methods with reused stages, showing that methods with reused stages and se effective stages are equivalent to three-step peer methods with se stages. Then, we analyze all the families of methods with two effective stages, obtaining methods with s=3 and orders 4 and 5 in which the free parameters of the families have been used to minimize the coefficient of the leading error term as well as to maximize the absolute stability interval. We have also studied one family of peer methods with s=4 and three effective stages, obtaining a method with order 6, superconvergent of order 7, and optimized leading error term as well as absolute stability interval. Some numerical experiments show the performance of the obtained methods by comparing them with other previously obtained peer methods as well as other standard Runge-Kutta and multistep methods.

Design of DIRK schemes with high weak stage order

Runge-Kutta (RK) methods may exhibit order reduction when applied to certain stiff problems. While fully implicit RK schemes exist that avoid order reduction via high-stage order, DIRK (diagonally implicit Runge-Kutta) schemes are practically important due to their structural simplicity; however, these cannot possess high stage order. The concept of weak stage order (WSO) can also overcome order reduction, and it is compatible with the DIRK structure. DIRK schemes of WSO up to 3 have been proposed in the past, however, based on a simplified framework that cannot be extended beyond WSO 3. In this work a general theory of WSO is employed to overcome the prior WSO barrier and to construct practically useful high-order DIRK schemes with WSO 4 and above. The resulting DIRK schemes are stiffly accurate, L-stable, have optimized error coefficients, and are demonstrated to perform well on a portfolio of relevant ODE and PDE test problems.

Explicit Nordsieck second derivative general linear methods for ODEs

The paper deals with the construction of explicit Nordsieck second derivative general linear methods with \(s\) stages of order \(p\) with \(p=s\) and high stage order \(q=p\) with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with \(r\) external stages and \(p=q=s=r-1\) up to order five are given, and numerical experiments in a fixed stepsize environment are presented. doi: https://doi.org/10.1017/S1446181122000049

EXPLICIT NORDSIECK SECOND DERIVATIVE GENERAL LINEAR METHODS FOR ODES

AbstractThe paper deals with the construction of explicit Nordsieck second derivative general linear methods with s stages of order p with $p=s$ and high stage order $q=p$ with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with r external stages and $p=q=s=r-1$ up to order five are given, and numerical experiments in a fixed stepsize environment are presented.

Implicit–explicit second derivative diagonally implicit multistage integration methods

We introduce a class of implicit–explicit (IMEX) schemes for the numerical solution of initial value problems of differential equations with both non-stiff and stiff components in which non-stiff and stiff solvers are, respectively, based on the explicit general linear methods (GLMs) and implicit second derivative GLMs (SGLMs). The order conditions of the proposed IMEX schemes are obtained. Linear stability properties of the methods are analyzed and then methods up to order four with a large area of absolute stability region of the pair are constructed assuming that the implicit part of the methods is L-stable. Due to the high stage orders of the constructed methods, they are not marred by order reduction. This is verified by the numerical experiments which demonstrate the efficiency of the proposed methods, too.

Convergence results for implicit–explicit general linear methods

This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.

A new class of efficient general linear methods for ordinary differential equations

We describe the construction of a new class of efficient general linear methods of high stage order, for nonstiff and stiff differential systems. Examples of explicit methods with large regions of stability, and implicit methods which are A- and L-stable, up to the order p=4 are presented. It is confirmed by numerical experiments that all methods achieve the expected order of accuracy and no order reduction occurs for stiff systems.

Alternating directions implicit integration in a general linear method framework

Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ADI methods have order at most two, due to the splitting errors. Moreover, when the time discretization of stiff one-dimensional problems is based on Runge–Kutta schemes, additional order reduction may occur. This work proposes a new ADI approach based on the partitioned General Linear Methods framework. This approach allows the construction of high order ADI methods. Due to their high stage order, the proposed methods can alleviate the order reduction phenomenon seen with other schemes. Numerical experiments are shown to provide further insight into the accuracy, stability, and applicability of these new methods.

A comparison of one-step and two-step W-methods and peer methods with approximate matrix factorization

One- and Two-step W-methods and peer methods are considered combined with an Approximate Matrix Factorization (AMF) for the numerical solution of large systems of stiff Ordinary Differential Equations (ODEs) coming from the spatial discretization of parabolic Partial Differential Equations (PDEs) of convection–diffusion–reaction type. Although one-step AMF W-methods do not require starting values and have larger stability regions, they may suffer from order reduction for such PDE problems when time-dependent Dirichlet boundary conditions are imposed. Here by numerical tests it is illustrated that two-step AMF W-methods and AMF peer methods avoid this order reduction (in stiff sense, i.e. for a fixed spatial grid) due to their high stage order. On the other hand, while AMF W-methods are linearly implicit, AMF peer methods require the solution of nonlinear equations in order to compute an advancing solution. In this case, the predictor considered for the internal stages influences both stability and accuracy. However, with only one Newton iteration, for a given number of stages, the three classes of methods have a similar computational cost.

General linear methods with large stability regions for Volterra integral equations

General linear methods are well-known as a large family of methods for the numerical solution of Volterra integral equations of the second kind. This paper is concerned with the construction of such methods with a large region of order p and high stage order $$q=p$$ with a large region of absolute stability. Some numerical results are presented which indicate the effectiveness of the proposed schemes.

Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

In this paper we study explicit peer methods with the strong stability preserving (SSP) property for the numerical solution of hyperbolic conservation laws in one space dimension. A system of ordinary differential equations is obtained by discontinuous Galerkin (DG) spatial discretizations, which are often used in the method of lines approach to solve hyperbolic differential equations. We present in this work the construction of explicit peer methods with stability regions that are designed for DG spatial discretizations and with large SSP coefficients. Methods of second- and third order with up to six stages are optimized with respect to both properties. The methods constructed are tested and compared with appropriate Runge–Kutta methods. The advantage of high stage order is verified numerically.

Construction of IMEX DIMSIMs of high order and stage order

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 analyze stability of these methods when the implicit and explicit parts interact with each other. We look for methods with large absolute stability region, assuming that the implicit part of the method is A(α)-, A-, or L-stable. Finally, we furnish examples of IMEX DIMSIMs of order p=5 and p=6 and stage order q=p, with good stability properties. Numerical examples illustrate that the IMEX schemes constructed in this paper do not suffer from order reduction phenomenon for some range of stepsizes.

How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems

The solution of stiff problems is frequently accompanied by a phenomenon known as order reduction. The reduction in the actual order can be avoided by applying methods with a fairly high stage order, ideally coinciding with the classical order. However, the stage order sometimes fails to be increased; moreover, this is not possible for explicit and diagonally implicit Runge–Kutta methods. An alternative approach is proposed that yields an effect similar to an increase in the stage order. New implicit and stabilized explicit Runge–Kutta methods are constructed that preserve their order when applied to stiff problems.

A new approach to the construction of DIMSIMs of high order and stage order

In this work we describe a new approach to the construction of diagonally implicit multistage integration methods (DIMSIMs) for the numerical solution of initial value problems for ordinary differential equations (ODEs). Differential Evolution, a very popular computational intelligence technique is employed to construct type 1 and type 2 methods with better or equivalent characteristics to the methods presented in the literature. The numerical results in selected problems justify this argument.

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.

Embedded symmetric nested implicit Runge–Kutta methods of Gauss and Lobatto types for solving stiff ordinary differential equations and Hamiltonian systems

A technique for constructing nested implicit Runge–Kutta methods in the class of mono-implicit formulas of this type is studied. These formulas are highly efficient in practice, since the dimension of the original system of differential equations is preserved, which is not possible in the case of implicit multistage Runge–Kutta formulas of the general from. On the other hand, nested implicit Runge-Kutta methods inherit all major properties of general formulas of this form, such as A-stability, symmetry, and symplecticity in a certain sense. Moreover, they can have sufficiently high stage and classical orders and, without requiring high extra costs, can ensure dense output of integration results of the same accuracy as the order of the underlying method. Thus, nested methods are efficient when applied to the numerical integration of differential equations of various sorts, including stiff and nonstiff problems, Hamiltonian systems, and invertible equations. In this paper, previously proposed nested methods based on the Gauss quadrature formulas are generalized to Lobatto-type methods. Additionally, a unified technique for constructing all such methods is proposed. Its performance is demonstrated as applied to embedded examples of nested implicit formulas of various orders. All the methods constructed are supplied with tools for local error estimation and automatic variable-stepsize mesh generation based on an optimal stepsize selection. These numerical methods are verified by solving test problems with known solutions. Additionally, a comparative analysis of these methods with Matlab built-in solvers is presented.

Rosenbrock methods with an explicit first stage

Traditional Rosenbrock methods suffer from order reduction when applied to partial differential equations with non-homogeneous boundary conditions and source terms. The paper studies a family of Rosenbrock schemes with an explicit first stage. This structure allows one to construct algorithms with high stage orders, and which do not suffer from order reduction. The paper discusses additional order conditions needed for linear stability, for using inexact Jacobians, and implementation aspects. Second- and third-order practical schemes are constructed, and their application to one- and two-dimensional partial differential equations test problems confirm the theoretical findings.

A Class Of Implicit-Explicit Two-Step Runge--Kutta Methods

This work develops new implicit-explicit time integrators based on two-step Runge--Kutta methods. The class of schemes of interest is characterized by linear invariant preservation and high stage orders. Theoretical consistency, stability, and stiff convergence analyses are performed to reveal the excellent properties of these methods. The new framework offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary. Practical schemes of orders three, four, and six are constructed and are used to solve several test problems. Numerical results confirm the theoretical findings.