scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

scholarly journals An Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

2019 ◽  
Vol 19 (2) ◽  
pp. 283-293 ◽  
Author(s):  
Vidar Thomée ◽  
A. S. Vasudeva Murthy
Keyword(s):  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Time Interval ◽  
Diffusion Term ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Strang Splitting ◽  
Spatially Periodic ◽  
Forward Euler

AbstractWe analyze a second-order accurate finite difference method for a spatially periodic convection-diffusion problem. The method is a time stepping method based on the Strang splitting of the spatially semidiscrete solution, in which the diffusion part uses the Crank–Nicolson method and the convection part the explicit forward Euler approximation on a shorter time interval. When the diffusion coefficient is small, the forward Euler method may be used also for the diffusion term.

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.

scholarly journals Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

2015 ◽  
Vol 25 (11) ◽  
pp. 2015-2042
Author(s):  
Erik Burman
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Burgers Equation ◽  
Euler Method ◽  
Second Order ◽  
Shock Capturing ◽  
Discrete Solution ◽  
A Priori Bound ◽  
Forward Euler ◽  
Forward Euler Method

We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory due to Tadmor. The estimates use a one-sided Lipschitz stability (Lip+-stability) estimate on the discrete solution and are obtained in a weak norm, but thanks to a total variation a priori bound on the discrete solution and an interpolation inequality, error estimates in Lp-norms (1 ≤ p < ∞) are deduced. Both first-order artificial viscosity and a nonlinear shock capturing term that formally is of second order are considered. For the discretization in time we use the forward Euler method. In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second-order strong stability preserving Runge–Kutta method. We also study the Lip+-stability property numerically and give some examples of when it holds strictly and when it is violated.

scholarly journals On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

2015 ◽  
Vol 52 (02) ◽  
pp. 323-338 ◽  
Author(s):  
Mihály Kovács ◽  
Stig Larsson ◽  
Fredrik Lindgren
Keyword(s):  
Gaussian Noise ◽  
Smooth Boundary ◽  
Spatial Domain ◽  
Euler Method ◽  
Euler Approximation ◽  
Implicit Euler Method ◽  
Cahn Equation ◽  
Backward Euler

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δt γ) for any γ &lt; ½. We also prove that the scheme converges uniformly in the strong L p -sense but with no rate given.

FINITE ELEMENT IMPLEMENTATION OF A SUPER-ELASTIC CONSTITUTIVE MODEL FOR TRANSFORMATION RATCHETTING OF NiTi ALLOY

2012 ◽  
Vol 09 (01) ◽  
pp. 1240022 ◽  
Author(s):  
Q. H. KAN ◽  
G. Z. KANG ◽  
S. J. GUO
Keyword(s):  
Finite Element ◽  
Constitutive Model ◽  
Niti Alloy ◽  
Cyclic Stress ◽  
Euler Method ◽  
Convergence Criterion ◽  
Integration Algorithm ◽  
Finite Element Implementation ◽  
Backward Euler ◽  
Forward Transformation

In the previous work, a new constitutive model describing the transformation ratchetting of super-elastic NiTi alloy was proposed. The finite element implementation of the proposed model is discussed in this work, because such implementation is necessary to launch a numerical analysis for the cyclic stress–strain responses of NiTi alloy devices including the transformation ratchetting. During the implementation, a new stress integration algorithm is adopted, and a new expression of the consistent tangent modulus is derived for the forward transformation and the reverse transformation. The finite element implementation is elaborated by the user subroutine of UMAT in ABAQUS based on backward Euler method. The accumulated error during cyclic transformation is controlled by a robust convergence criterion. Finally, the validity of such implementation is verified by several numerical examples.

scholarly journals Development of a continuum plasticity model for the commercial finite element code ABAQUS

2011 ◽  
Vol 2 (2) ◽  
pp. 275-283
Author(s):  
M. Safaei ◽  
W. De Waele
Keyword(s):  
Finite Element ◽  
Yield Surface ◽  
Euler Method ◽  
Integration Scheme ◽  
Isotropic Hardening ◽  
Implicit Integration ◽  
Backward Euler ◽  
User Material Subroutine ◽  
Von Mises ◽  
Future Work

The present work relates to the development of computational material models for sheet metalforming simulations. In this specific study, an implicit scheme with consistent Jacobian is used forintegration of large deformation formulation and plane stress elements. As a privilege to the explicitscheme, the implicit integration scheme is unconditionally stable. The backward Euler method is used toupdate trial stress values lying outside the yield surface by correcting them back to the yield surface atevery time increment. In this study, the implicit integration of isotropic hardening with the von Mises yieldcriterion is discussed in detail. In future work it will be implemented into the commercial finite element codeABAQUS by means of a user material subroutine.

scholarly journals The Sdfem for a Convection-diffusion Problem with Two Small Parameters

2003 ◽  
Vol 3 (3) ◽  
pp. 443-458 ◽  
Author(s):  
Hans-Görg Roos ◽  
Zorica Uzelac
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Small Parameters ◽  
Perturbation Parameters ◽  
Theoretical Results

AbstractA singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

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