Bi-discontinuous time step integration algorithms––Part 1: first order equations

2003 ◽  
Vol 192 (3-4) ◽  
pp. 331-349 ◽  
Author(s):  
T.C. Fung
Keyword(s):  
Time Step ◽  
First Order ◽  
Integration Algorithms

scholarly journals A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 769-782
Author(s):  
Amiya K. Pani ◽  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Second Order ◽  
Time Step ◽  
Convection Diffusion ◽  
First Order ◽  
Backward Euler ◽  
Strang Splitting ◽  
Spatially Periodic

AbstractWe analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.

Complex network structure of flocks in the Vicsek Model with Vectorial Noise

2014 ◽  
Vol 25 (03) ◽  
pp. 1350095 ◽  
Author(s):  
Gabriel Baglietto ◽  
Ezequiel V. Albano ◽  
Julián Candia
Keyword(s):  
Phase Transition ◽  
Complex Network ◽  
Network Structure ◽  
Time Step ◽  
Vicsek Model ◽  
Disorder Phase Transition ◽  
First Order ◽  
Direction Of Movement ◽  
Long Range Ordering ◽  
Influence Of Noise

In the Vicsek Model (VM), self-driven individuals try to adopt the direction of movement of their neighbors under the influence of noise, thus leading to a noise-driven order–disorder phase transition. By implementing the so-called Vectorial Noise (VN) variant of the VM (i.e. the VM-VN model), this phase transition has been shown to be discontinuous (first-order). In this paper, we perform an extensive complex network study of VM-VN flocks and show that their topology can be described as highly clustered, assortative, and nonhierarchical. We also study the behavior of the VM-VN model in the case of "frozen flocks" in which, after the flocks are formed using the full dynamics, particle displacements are suppressed (i.e. only rotations are allowed). Under this kind of restricted dynamics, we show that VM-VN flocks are unable to support the ordered phase. Therefore, we conclude that the particle displacements at every time-step in the VM-VN dynamics are a key element needed to sustain long-range ordering throughout.

Comparison of Two Time-Integration Algorithms for an Anisotropic Damage Model Coupled with Plasticity

2015 ◽  
Vol 784 ◽  
pp. 292-299 ◽  
Author(s):  
Stephan Wulfinghoff ◽  
Marek Fassin ◽  
Stefanie Reese
Keyword(s):  
Evolution Equations ◽  
Damage Model ◽  
Time Integration ◽  
Three Dimensional ◽  
Anisotropic Damage ◽  
Euler Scheme ◽  
The Finite Element Method ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Integration Algorithms

In this work, two time integration algorithms for the anisotropic damage model proposed by Lemaitre et al. (2000) are compared. Specifically, the standard implicit Euler scheme is compared to an algorithm which implicitly solves the elasto-plastic evolution equations and explicitly computes the damage update. To this end, a three dimensional bending example is solved using the finite element method and the results of the two algorithms are compared for different time step sizes.

Construction of Higher-Order Accurate Time-Step Integration Algorithms by Equal-Order Polynomial Projection

2005 ◽  
Vol 11 (1) ◽  
pp. 19-49 ◽  
Author(s):  
T. C. Fung
Keyword(s):  
Integral Relation ◽  
Higher Order ◽  
Numerical Results ◽  
Numerical Dissipation ◽  
Time Interval ◽  
Time Step ◽  
Higher Order Terms ◽  
Optimal Polynomial ◽  
Integration Algorithms ◽  
New Framework

This paper presents a new framework to construct higher-order accurate time-step integration algorithms based on weakly enforcing the differential/integral relation. The dependent variable and its time derivatives are assumed to be polynomials of equal order. A differential equation is then transformed into an algebraic equation directly. The main issue is how to approximate the integral of a polynomial by another polynomial of the same degree. Various methods to determine the optimal representation (or projection) are considered. It is shown that to reproduce numerical results equivalent to the Padé or generalized Padé approximations, the coefficients of the optimal polynomial representation are related to the weighting parameters derived previously for time-step integration algorithms with predetermined coefficients. A special feature for the present formulation is that the same procedure can be used to solve first-, second-, and even higher-order non-homogeneous initial value problems in a unified manner. The resultant algorithms are higher-order accurate and unconditionally A-stable with controllable numerical dissipation. It is also shown that for the numerical results to maintain higher-order accuracy at the end of a time interval, the higher-order terms in the excitation have to be projected as polynomials of lower degree within the present framework as well. Numerical examples are given to illustrate the validity of the present formulations.

scholarly journals Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Reconstructions for Forward-in-Time Schemes

2010 ◽  
Vol 138 (12) ◽  
pp. 4497-4508 ◽  
Author(s):  
William C. Skamarock ◽  
Maximo Menchaca
Keyword(s):  
Volume Transport ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
First Order ◽  
Order Reconstruction ◽  
Flow Test ◽  
Mesh Density ◽  
Original Scheme ◽  
Dependent Flow

Abstract The finite-volume transport scheme of Miura, for icosahedral–hexagonal meshes on the sphere, is extended by using higher-order reconstructions of the transported scalar within the formulation. The use of second- and fourth-order reconstructions, in contrast to the first-order reconstruction used in the original scheme, results in significantly more accurate solutions at a given mesh density, and better phase and amplitude error characteristics in standard transport tests. The schemes using the higher-order reconstructions also exhibit much less dependence of the solution error on the time step compared to the original formulation. The original scheme of Miura was only tested using a nondeformational time-independent flow. The deformational time-dependent flow test used to examine 2D planar transport in Blossey and Durran is adapted to the sphere, and the schemes are subjected to this test. The results largely confirm those generated using the simpler tests. The results also indicate that the scheme using the second-order reconstruction is most efficient and its use is recommended over the scheme using the first-order reconstruction. The second-order reconstruction uses the same computational stencil as the first-order reconstruction and thus does not create any additional parallelization issues.

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