A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

AbstractSince integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.

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Some notes on summation by parts time integration methods

Results in Applied Mathematics ◽

10.1016/j.rinam.2019.100004 ◽

2019 ◽

Vol 1 ◽

pp. 100004 ◽

Cited By ~ 7

Author(s):

Hendrik Ranocha

Keyword(s):

Time Integration ◽

Integration Methods ◽

Summation By Parts ◽

Time Integration Methods

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A New Class of Higher Order Accurate Non-Oscillatory Implicit Time Integration Schemes

8th AIAA/CEAS Aeroacoustics Conference & Exhibit ◽

10.2514/6.2002-2510 ◽

2002 ◽

Author(s):

Karthikeyan Duraisamy ◽

James Baeder

Keyword(s):

Time Integration ◽

Higher Order ◽

Implicit Time ◽

New Class ◽

Integration Schemes ◽

Implicit Time Integration ◽

Time Integration Schemes

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Implementation of multirate time integration methods for air pollution modelling

Geoscientific Model Development ◽

10.5194/gmd-5-1395-2012 ◽

2012 ◽

Vol 5 (6) ◽

pp. 1395-1405 ◽

Cited By ~ 7

Author(s):

M. Schlegel ◽

O. Knoth ◽

M. Arnold ◽

R. Wolke

Keyword(s):

Time Integration ◽

Efficient Implementation ◽

Step Size ◽

Time Step ◽

Worst Case ◽

Integration Methods ◽

Integration Schemes ◽

Numerical Effort ◽

Multirate Time Integration ◽

Time Integration Methods

Abstract. Explicit time integration methods are characterised by a small numerical effort per time step. In the application to multiscale problems in atmospheric modelling, this benefit is often more than compensated by stability problems and step size restrictions resulting from stiff chemical reaction terms and from a locally varying Courant-Friedrichs-Lewy (CFL) condition for the advection terms. Splitting methods may be applied to efficiently combine implicit and explicit methods (IMEX splitting). Complementarily multirate time integration schemes allow for a local adaptation of the time step size to the grid size. In combination, these approaches lead to schemes which are efficient in terms of evaluations of the right-hand side. Special challenges arise when these methods are to be implemented. For an efficient implementation, it is crucial to locate and exploit redundancies. Furthermore, the more complex programme flow may lead to computational overhead which, in the worst case, more than compensates the theoretical gain in efficiency. We present a general splitting approach which allows both for IMEX splittings and for local time step adaptation. The main focus is on an efficient implementation of this approach for parallel computation on computer clusters.

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A weak form temporal quadrature element formulation for linear structural dynamics

Engineering Computations ◽

10.1108/ec-07-2020-0377 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Junning Qin ◽

Hongzhi Zhong

Keyword(s):

Structural Dynamics ◽

Initial Conditions ◽

Time Integration ◽

Weak Form ◽

Element Formulation ◽

Dynamic Problems ◽

Structural Dynamic ◽

Integration Methods ◽

Content Type ◽

Time Integration Methods

PurposeVarious time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.Design/methodology/approachA variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.FindingsThe weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.Originality/valueThe weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.

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