Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Hundsdorfer, Willem; Steininger, B.I.

doi:10.1007/bf01952789

Cited by 19 publications

(20 citation statements)

References 17 publications

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“…We shall diHcuss the accuracy of the schemes with variable entries in some detail in section 5, since the standard local truncation error no longer gives proper information about the accuracy of these schemes. This i:; similar to the situation for :;tiff ODEs as considered in Hundsdorfer and Steininger [12]. Numerical results will be presented in section 6 for a test example from reservoir simulation, where we have locally large convective velocities q near injection and production wells and moderate or small velocities elsewhere in the spatial region.…”

Section: 2) W'(t) = F(tw(t)) T 2 0mentioning

confidence: 61%

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Partially Implicit BDF2 Blends for Convection Dominated Flows

Hundsdorfer¹

2001

SIAM J. Numer. Anal.

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Abstract. In this paper we consider various blends of implicit and explicit time integration schemes, based on the well-known BDF2 met.hod, applied to convection-diffusion problems with dominating convection. A fully implicit treatment of convection terms is often not very efficient. We shall deal with second order schemes that are implicit in the convection terms only locally in space, without introducing the internal inconsistencies that are common with many time-splitting methods. Along with implementation aspects of the implicit relations, we shall discuss accuracy of the schemes, positivity and monotonicity properties.

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Section: 2) W'(t) = F(tw(t)) T 2 0mentioning

confidence: 61%

“…Thus one might expect the global errors to be first order only. However, smnl~r to [12] and [.10, sect. V.7] for stiff ODEs.…”

Section: Global Accuracy Resultsmentioning

confidence: 99%

Partially Implicit BDF2 Blends for Convection Dominated Flows

Hundsdorfer¹

2001

SIAM J. Numer. Anal.

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“…The implicit midpoint rule can also be regarded as a one-leg multistep method. Most one-leg methods have p = q + 1 and for these methods condition (4.5) is always fulfilled, as can be seen from the convergence results in Hundsdorfer & Steininger (1991) and Hairer & Wanner (1991) (or by noting that the functions <p and <P are identical if k = 1), but algebraically stable (G-stable) one-leg multistep methods have p 3= 2.…”

Section: Non-linear Problemsmentioning

confidence: 94%

“…Convergence with order q + 1 follows easily by considering the modified errors e H = e n -xh" + y +l \t n ) (4.6) which satisfy a recursion with C, only depending on derivatives of y, see for instance Burrage & Hundsdorfer (1987), Hundsdorfer & Steininger (1991).…”

Section: Non-linear Problemsmentioning

confidence: 99%

On the error of general linear methods for stiff dissipative differential equations

Hundsdorfer

1994

IMA J Numer Anal

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Many numerical methods to solve initial value problems of the form y' =f(t, y) can be written as general linear methods. Classical convergence results for such methods are based on the Lipschitz constant and bounds for certain partial derivatives of /. For stiff problems these quantities may be very large, and consequently the classical order of convergence loses its significance. In this paper we consider bounds for the global errors which are based only on bounds for derivatives of y for linear and non-linear dissipative problems with arbitrary stiffness.

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“…Rigorous proofs are presented for the cross-diffusion system (11)- (12) in Section IVB and for the fourth-order quantum diffusion Eq. (18) in Section IVC.…”

Section: Existence Of Semi-discrete Solutionsmentioning

confidence: 99%

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

Jüngel¹,

Milišić

2014

Numerical Methods Partial

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New one-leg multistep time discretizations of nonlinear evolution equations are investigated. The main features of the scheme are the preservation of the non-negativity and the entropy dissipation structure of the diffusive equations. The key ideas are to combine Dahlquist's G-stability theory with entropy dissipation methods and to introduce a nonlinear transformation of variables, which provides a quadratic structure in the equations. It is shown that G-stability of the one-leg scheme is sufficient to derive discrete entropy dissipation estimates. The general result is applied to a cross-diffusion system from population dynamics and a nonlinear fourth-order quantum diffusion model, for which the existence of semidiscrete weak solutions is proved. Under some assumptions on the operator of the evolution equation, the second-order convergence of solutions is shown. Moreover, some numerical experiments for the population model are presented, which underline the theoretical results.

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Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

Cited by 19 publications

References 17 publications

Partially Implicit BDF2 Blends for Convection Dominated Flows

Partially Implicit BDF2 Blends for Convection Dominated Flows

On the error of general linear methods for stiff dissipative differential equations

Entropy dissipative one‐leg multistep time approximations of nonlinear diffusive equations

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