Convergence results of two-step W-methods for two-parameter singular perturbation problems

doi:10.1016/j.amc.2006.11.131

Applied Mathematics and Computation

Volume 189, Issue 1, 1 June 2007, Pages 669-681

https://doi.org/10.1016/j.amc.2006.11.131 Get rights and content

Abstract

Two-step W-methods are a class of efficient numerical methods for stiff initial value problems of ordinary differential equations. We study quantitative convergence of parallel two-step W-methods for a class of two-parameter singular perturbation problems, obtain the local and global error estimates for variable stepsizes, show that no order reduction occurs, and extend the corresponding results given by Weiner et al. [R. Weiner, B.A. Schmitt, H. Podhaisky, Two-step W-methods on singular perturbation problems, Report 73, FB Mathematik und Informatik, Universität Marburg, Marburg, 2000].

Introduction

The singular-perturbation initial value problems arise widely in many scientific and engineering fields such as chemical kinetics, automatics control, fluid mechanics, etc. They are a special class of stiff initial value problems. But they cannot be covered by B-theory (cf. [1], [2], [3], [4], [5], [6], [7], [8]), due to their one-sided Lipschitz constants which are in general of size O(ε⁻¹) with 0 < ε ≪ 1.

In the last ten years, many authors have presented many important convergence results for linear multistep methods, Runge–Kutta methods, Rosenbrock methods, one-leg methods, general linear methods, etc., for one-parameter singular perturbation problems (SPPs) (cf. [9], [10], [11], [12], [13], [14], [15], [16]). It is more complicated and more difficult to discuss the error behaviors of numerical methods for SPPs with many parameters being of multiple scales than with one parameter. So far, there are a few convergence results of A(α)-stable linear multistep methods for two-parameter SPPs (cf. [17]).

Recently, linearly-implicit two-step methods were introduced and widely investigated for solving stiff initial value problems (cf. [18], [19], [20], [21], [22], [23], [24]). These methods include parallel two-step W-methods (PTSW methods) (cf. [18], [19], [20]) and parallel peer two-step methods (cf. [21], [22], [23], [24]), etc. These parallel methods overcome some drawbacks of linearly-implicit one-step methods and have some good characterizations, which are summed up in [24]. Especially, convergence properties of PTSW methods applied to one-parameter SPPs were discussed, and estimates of the global error for variable stepsizes are given in [18]. Convergence properties of a class of implicit parallel peer two-step methods for one-parameter SPPs were also investigated for constant stepsize in [23]. The main results in [18], [23] are that no order reduction occurs.

In the present paper, we discuss quantitative error behaviors of PTSW methods for a class of SPPs with two-parameter. These extend the corresponding results described in [18], and also show that no order reduction occurs.

Section snippets

Two-parameter SPPs and PTSW methods

Consider the two-parameter singularly-perturbed initial value problem $\{\begin{matrix} x^{'} (t) = u (x, y, z), x \in [t_{0}, T], \\ ε_{1} y^{'} (t) = f (x, y, z), \\ ε_{2} z^{'} (t) = g (x, y, z), \\ x (t_{0}) = x_{0}, y (t_{0}) = y_{0}, z (t_{0}) = z_{0}, \end{matrix}$ where ε₁, ε₂ are perturbation parameters and satisfy 0 < ε_i ≪ 1 for i = 1, 2. Maps $u : R^{n_{x}} \times R^{n_{y}} \times R^{n_{z}} \to R^{n_{x}}$ , $f : R^{n_{x}} \times R^{n_{y}} \times R^{n_{z}} \to R^{n_{y}}$ and $g : R^{n_{x}} \times R^{n_{y}} \times R^{n_{z}} \to R^{n_{z}}$ are sufficiently smooth with bounded derivatives, and the functions f, g satisfy $\begin{matrix} μ (f_{y} (x, y, z)) ⩽ μ_{1} < 0, \\ μ (g_{z} (x, y, z)) ⩽ μ_{2} < 0, \forall x \in R^{n_{x}}, y \in R^{n_{y}}, z \in R^{n_{z}}, \end{matrix}$ where μ(·) denotes the logarithmic matrix norm with respect to the Euclidean

Simplifying conditions and local error

Now, we consider the residual errors $δ_{mi}$ and $η_{m}$ that appear when the exact values $w (t_{m})$ and $w^{'} (t_{m} + c_{i} h_{m})$ are substituted for w_m and k_mi in (2.4) $\{\begin{matrix} {\hat{W}}_{mi} = w (t_{m}) + h_{m} \sum_{j = 1}^{s} a_{ij} w^{'} (t_{m - 1} + c_{j} h_{m - 1}), i = 1, \dots, s, \\ (I - h_{m} γ_{i} T_{m}) w^{'} (t_{m} + c_{i} h_{m}) = F ({\hat{W}}_{mi}) + h_{m} T_{m} \sum_{j = 1}^{s} γ_{ij} w^{'} (t_{m - 1} + c_{j} h_{m - 1}) + δ_{mi}, \\ w (t_{m + 1}) = w (t_{m}) + h_{m} \sum_{i = 1}^{s} (b_{i} w^{'} (t_{m} + c_{i} h_{m}) + v_{i} w^{'} (t_{m - 1} + c_{i} h_{m - 1})) + η_{m} . \end{matrix}$

The errors in (3.1) can be discussed with the help of the following simplifying conditions (cf. [19]): $\begin{matrix} C (q) : \sum_{j = 1}^{s} a_{ij} (c_{j} - 1)^{l - 1} = σ^{l - 1} \frac{c_{i}^{l}}{l}, l = 1, 2, \dots, q, \\ Γ (q) : \sum_{j = 1}^{s} γ_{ij} (c_{j} - 1)^{l - 1} = - γ_{i} σ^{l - 1} c_{i}^{l - 1}, l = 1, 2, \dots, q, \end{matrix}$ $B ($

PTSW methods

To prove the main results, we introduce the following lemmas. The proof of these lemmas are similar to the corresponding lemmas given in [18]. In this section, we will study the quantitative errors of PTSW methods with $T_{m} = F^{'} (w_{m})$ .

Lemma 3

Assume that (2.5) and $f_{z} = 0$ hold, $T_{m}$ has the form (3.3), then $\begin{matrix} (I - h_{m} γ T_{m})^{- 1} \\ = (\begin{matrix} I + O (h_{m}) & O (ε_{1}) & O (ε_{2}) \\ O (1) & {(I - \frac{h_{m} γ}{ε_{1}} T_{m, 5})}^{- 1} + O (ε_{1}) & O (ε_{2}) \\ O (1) & O (\frac{ε_{1}}{h_{m}}) + O (ε_{1}) & {(I - \frac{h_{m} γ}{ε_{2}} T_{m, 9})}^{- 1} + O (ε_{2}) \end{matrix}), \end{matrix}$ $\begin{matrix} S_{m} = (I - h_{m} γ T_{m})^{- 1} h_{m} T_{m} \\ = (\begin{matrix} O (h_{m}) & O (ε_{1}) & O (ε_{2}) \\ O (1) & {(I - \frac{h_{m} γ}{ε_{1}} T_{m, 5})}^{- 1} \frac{h_{m}}{ε_{1}} T_{m, 5} + O (ε_{1}) & O (ε_{2}) \\ O (1) & O (\frac{ε_{1}}{h_{m}}) + O (ε_{1}) & {(I - \frac{h_{m} γ}{ε_{2}} T_{m, 9})}^{- 1} \frac{h_{m}}{ε_{2}} T_{m, 9} + O (ε_{2}) \end{matrix}) . \end{matrix}$

Proof

Partitioned PTSW methods

We will discuss global errors of the partitioned W-method in this section. The difference between this method and the PSTW method which we considered in the above section is the choice of T_m. Here, we suppose that $T_{m} = P^{- 1} (\begin{matrix} 0 & 0 & 0 \\ T_{m, 4} & T_{m, 5} & T_{m, 6} \\ T_{m, 7} & T_{m, 8} & T_{m, 9} \end{matrix}) .$

In this situation, the component of numerical solution w_m which approximates to $x (t_{m})$ can be computed explicitly. So the size of the linear system of the method (2.4) reduces from n to $n - n_{x}$ .

With the assumption (5.1), it is straightforward to show that

Numerical examples

Consider the two-stage PTSW method (cf. [18]) $\begin{matrix} γ = 1.2, c_{1} = 1 + \frac{γ - \frac{1}{3}}{γ - \frac{1}{2}}, c_{2} = 1, \\ a_{11} = \frac{c_{1}^{2}}{2 (c_{1} - 1)}, a_{21} = \frac{c_{2}^{2}}{2 (c_{1} - 1)}, a_{12} = c_{1} - a_{11}, a_{22} = c_{2} - a_{21}, \\ γ_{11} = \frac{- γ c_{1}}{c_{1} - 1}, γ_{21} = \frac{- γ c_{2}}{c_{1} - 1}, γ_{12} = - γ - γ_{11}, γ_{22} = - γ - γ_{21}, \\ b_{1} = \frac{2 c_{1} - 3}{12 c_{1} (c_{1} - 1)}, b_{2} = \frac{6 c_{1}^{2} - 14 c_{1} + 7}{12 (c_{1} - 2) (c_{1} - 1)}, \\ v_{1} = - \frac{2 c_{1} - 1}{12 (c_{1} - 2) (c_{1} - 1)}, v_{2} = \frac{6 c_{1}^{2} - 10 c_{1} + 3}{12 c_{1} (c_{1} - 1)} . \end{matrix}$

This method satisfies B(4), C(2) and $Γ (2)$ . It is a method of order 3 according to Theorem 1, Theorem 2. We apply this method to the following examples.

Problem 1

Consider the nonlinear problem $\{\begin{matrix} x^{'} (t) = u (x, y, z) = x - (t - 1) y + z - \sin t, t \in [0, 1], \\ ε_{1} y^{'} (t) = f (x, y \end{matrix}$

Acknowledgements

The authors are indebted to the referees for their useful comments and valuable suggestions.

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^☆: This work is supported by project from NSF of China (No.10571147).

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Convergence results of two-step W-methods for two-parameter singular perturbation problems☆

Abstract

Introduction

Section snippets

Two-parameter SPPs and PTSW methods

Simplifying conditions and local error

PTSW methods

Partitioned PTSW methods

Numerical examples

Acknowledgements

Appl. Numer. Math.

Appl. Numer. Math.

Appl. Numer. Math.

Appl. Numer. Math.

A stability property of implicit Runge–Kutta methods

BIT

Stability criteria for implicit Runge–Kutta methods

SIAM J. Numer. Anal

Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations

Stability properties implicit Runge–Kutta methods

SIAM J. Numer. Anal.

Order results for implicit Runge–Kutta methods applied to stiff systems

SIAM J. Numer. Anal.

Theory of Computational Methods for Stiff Differential Equations

Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems