Ordinary Differential Equations in Theory and Practice

Mattheij, R.M.M.; Molenaar, Jaap

doi:10.1137/1.9780898719178

Cited by 57 publications

(62 citation statements)

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“…e.g. [10]) that the local discretisation error of the Euler Backward method applied to (2.1) is given by…”

Section: Error Analysismentioning

confidence: 99%

“…(see [10]). From this it is clear that interpolation may be the dominant error source and thus determine the accuracy of the method.…”

Section: Error Analysismentioning

confidence: 99%

“…Of course, in the general (nonlinear) case we cannot say that the stability properties of the flow method are the same as those of the Euler Backward method. Hence, we need a different approach (see [1], [10]) to investigate its stability. We can formulate our problem as a final solution of the one step method…”

Section: Stabilitymentioning

confidence: 99%

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An explicit method for solving flows of ODE

Tasic

Mattheij

2004

Applied Mathematics and Computation

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This paper is concerned with finding numerical solutions of a flow of ODE solutions. It describes a new method, based on Euler Backward method and interpolation, for finding solutions of autonomous problems. The special aspect is that this method is explicit, and resolves the flow with similar accuracy as Euler Backward method, even when the problem is stiff. An analysis is given of both stability and accuracy and extensions to non-autonomous problems are discussed. A number of numerical examples illustrates this analysis.

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“…e.g. [10]) that the local discretisation error of the Euler Backward method applied to (2.1) is given by…”

Section: Error Analysismentioning

confidence: 99%

“…(see [10]). From this it is clear that interpolation may be the dominant error source and thus determine the accuracy of the method.…”

Section: Error Analysismentioning

confidence: 99%

See 1 more Smart Citation

An explicit method for solving flows of ODE

Tasic

Mattheij

2004

Applied Mathematics and Computation

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“…Note that the differential index of a system of ODEs is 0. For a thorough discussion of differential-algebraic equations refer to [1,16].…”

Section: Time Continuous Governing Equationsmentioning

confidence: 99%

On dual Schur domain decomposition method for linear first-order transient problems

Nakshatrala¹,

Prakash²,

Hjelmstad³

2009

Journal of Computational Physics

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Abstract. This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. The spatially discrete system of equations resulting from a dual Schur domain decomposition method can be expressed as a system of differential algebraic equations (DAEs). In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Unlike the continuous formulation, the discretized formulation of the transient problem is unable to enforce simultaneously the continuity of both the primary variable and its rate along the subdomain interface (except for the backward Euler method).Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the "energy" method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs). Finally, we present numerical examples to corroborate our theoretical predictions.

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“…It is well known (see e.g. [12]) that EB has consistency order 1 and IMR order 2, i.e. the error d is O(h) and O(h 2 ) respectively.…”

Section: Error Analysismentioning

confidence: 99%

Explicitly solving vectorial ODEs

Tasic

Mattheij

2005

Applied Mathematics and Computation

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This paper concerns a new method for finding numerical solutions of multivariate ODE flows, where the flow field is not given explicitly. The method is based on the existing implicit numerical methods, such as Euler Backward and the implicit midpoint rule. The main aspect is that the method as such is explicit, but with the similar accuracy and stability properties as the original (implicit) methods. A higher dimensional analysis is given of both stability and accuracy. A couple of examples illustrates this analysis.

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Ordinary Differential Equations in Theory and Practice

Cited by 57 publications

References 0 publications

An explicit method for solving flows of ODE

An explicit method for solving flows of ODE

On dual Schur domain decomposition method for linear first-order transient problems

Explicitly solving vectorial ODEs

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