A class of half-explicit Runge-Kutta methods for differential-algebraic systems of index 3

Ostermann, Alexander

doi:10.1016/0168-9274(93)90140-m

Cited by 14 publications

(8 citation statements)

References 3 publications

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“…However, it is not obvious that the choice (b i ) = (b i ) and (B i ) = (B i ) implies that (4) has the same order as (3). In fact, according to results given by Ostermann [12] for semi-explicit methods for DAEs of index three, one has to expect order reduction, in general. This will be made explicit for (4) in Sect.…”

Section: Semi-explicit Prk Methods For Constrained Systemsmentioning

confidence: 99%

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On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

Sebastian

1997

Numerische Mathematik

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In this paper we generalize the class of explicit partitioned RungeKutta (PRK) methods for separable Hamiltonian systems to systems with holonomic constraints. For a convenient analysis of such schemes, we first generalize the backward error analysis for systems in R m to systems on manifolds embedded in R m . By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suffer from order reduction as well-known for higher-index differential-algebraic equations. However, this order reduction can be avoided by a proper modification of the standard PRK methods. This modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant. We also give a numerical comparison of several second, fourth, and sixth order methods. Classification (1991): 65L05, 65L20 Mathematics Subject

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Section: Semi-explicit Prk Methods For Constrained Systemsmentioning

confidence: 99%

“…Let X and Y be two vector fields on M with corresponding flows exp (tX ) and exp (tY ) respectively. Then there is a perturbed vector fieldẐ (h) such that exp (hẐ (h)) = exp (hY ) · exp (hX ) (12) up to terms of arbitrarily high order. The expansion ofẐ (h) is given bŷ…”

Section: Backward Error Analysis For Integrators On Manifoldsmentioning

confidence: 99%

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

Sebastian

1997

Numerische Mathematik

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“…To be precise, the DAE system is in semi-explicit form and has Hessenberg index 3 2 . The existence of solutions for our index 3 problem is guaranteed since F I is linear in p [40].…”

Section: Derivation Of the Index And Hidden Constraintsmentioning

confidence: 99%

Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow

Sanderse

Veldman

2019

Journal of Computational Physics

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New time integration methods are proposed for simulating incompressible multiphase flow in pipelines described by the one-dimensional two-fluid model. The methodology is based on 'halfexplicit' Runge-Kutta methods, being explicit for the mass and momentum equations and implicit for the volume constraint. These half-explicit methods are constraint-consistent, i.e., they satisfy the hidden constraints of the two-fluid model, namely the volumetric flow (incompressibility) constraint and the Poisson equation for the pressure. A novel analysis shows that these hidden constraints are present in the continuous, semi-discrete, and fully discrete equations.Next to constraint-consistency, the new methods are conservative: the original mass and momentum equations are solved, and the proper shock conditions are satisfied; efficient: the implicit constraint is rewritten into a pressure Poisson equation, and the time step for the explicit part is restricted by a CFL condition based on the convective wave speeds; and accurate: achieving high order temporal accuracy for all solution components (masses, velocities, and pressure). Highorder accuracy is obtained by constructing a new third-order Runge-Kutta method that satisfies the additional order conditions arising from the presence of the constraint in combination with time-dependent boundary conditions. Two test cases (Kelvin-Helmholtz instabilities in a pipeline and liquid sloshing in a cylindrical tank) show that for time-independent boundary conditions the half-explicit formulation with a classic fourth-order Runge-Kutta method accurately integrates the two-fluid model equations in time while preserving all constraints. A third test case (ramp-up of gas production in a multiphase pipeline) shows that our new third-order method is preferred for cases featuring time-dependent boundary conditions.

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“…These concepts are standard in DAE theory, cf. [3,6,8,9,26,35]. On the other hand, we follow the taming or truncation approach developed in recent years in the context of numerical methods for SDEs with non-globally Lipschitz continuous coefficients, cf.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

Lindner¹,

Stroot²

2017

Preprint

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This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally onesided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform Lp-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semi-stability and moment growth properties of the scheme. To the best of our knowledge, the presented result is the first strong convergence result for a constraint-preserving scheme in the considered setting.

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A class of half-explicit Runge-Kutta methods for differential-algebraic systems of index 3

Cited by 14 publications

References 3 publications

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

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