Time Discretization for Capillary Flow: Beyond Backward Euler

2017 ◽  
pp. 121-143
Author(s):  
Stephan Weller ◽  
Eberhard Bänsch
Keyword(s):  
Capillary Flow ◽  
Time Discretization ◽  
Backward Euler

scholarly journals A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

2020 ◽  
Author(s):  
Sergio Caucao ◽  
Ivan Yotov
Keyword(s):  
Banach Space ◽  
Time Discretization ◽  
Monotone Operators ◽  
Rates Of Convergence ◽  
Mixed Formulation ◽  
Model Parameters ◽  
Weak Formulation ◽  
Finite Element Approximations ◽  
Backward Euler ◽  
Discrete Finite Element

Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

scholarly journals Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

2014 ◽  
Vol 14 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Dominik Meidner ◽  
Thomas Richter
Keyword(s):  
Time Discretization ◽  
Posteriori Error ◽  
Numerical Dissipation ◽  
Error Estimator ◽  
Fractional Step ◽  
Weighted Residual Method ◽  
Time Stepping ◽  
Backward Euler ◽  
Galerkin Formulation ◽  
Theta Method

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

2020 ◽  
Vol 20 (2) ◽  
pp. 251-272
Author(s):  
Christoph Erath ◽  
Robert Schorr
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Time Discretization ◽  
Euler Method ◽  
Interface Problems ◽  
Backward Euler Method ◽  
Elliptic Interface Problems ◽  
Practical Applications ◽  
Backward Euler ◽  
Volume Method

AbstractMany problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

Stability and consistency of a finite difference scheme for compressible viscous isentropic flow in multi-dimension

2018 ◽  
Vol 26 (3) ◽  
pp. 111-140 ◽  
Author(s):  
Radim Hošek ◽  
Bangwei She
Keyword(s):  
Time Discretization ◽  
Euler Method ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Numerical Density ◽  
Existence Of A Solution ◽  
Isentropic Flow ◽  
Backward Euler ◽  
Spatial Dimensions ◽  
Stability And Consistency

Abstract Motivated by the work of Karper [29], we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly positivity of the numerical density, stability and consistency of the method for the whole range of physically relevant adiabatic exponents are proved. The theoretical part is complemented by computational results that are performed in two spatial dimensions.

scholarly journals On the locking-free three-field virtual element methods for Biot's consolidation model in poroelasticity

2020 ◽  
Author(s):  
Xialan Tang ◽  
Zhibin Liu ◽  
Baiju Zhang ◽  
Minfu Feng
Keyword(s):  
Time Discretization ◽  
Optimal Error Estimates ◽  
Specific Storage ◽  
High Order Scheme ◽  
Backward Euler ◽  
Consolidation Model ◽  
Order Scheme ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Biot’S Consolidation

We propose and analyze two locking-free three-field virtual element methods for Biot's consolidation model in poroelasticity. One is a high-order scheme, and the other is a low-order scheme. For time discretization, we use the backward Euler scheme. The proposed methods are well-posed, and optimal error estimates of all the unknowns are obtained for fully discrete solutions. The generic constants in the estimates are uniformly bounded as the Lamé coefficient λ tends to infinity, and as the constrained specific storage coefficient is arbitrarily small. Therefore the methods are free of both Poisson locking and pressure oscillations. Numerical results illustrate the good performance of the methods and confirm our theoretical predictions.

scholarly journals Optimizing the Kelvin force in a moving target subdomain

2017 ◽  
Vol 28 (01) ◽  
pp. 95-130 ◽  
Author(s):  
Harbir Antil ◽  
Ricardo H. Nochetto ◽  
Pablo Venegas
Keyword(s):  
Magnetic Field ◽  
Time Discretization ◽  
Dipole Approximation ◽  
Second Order ◽  
Sufficient Condition ◽  
Final Time ◽  
Magnetic Drug Delivery ◽  
Backward Euler ◽  
Pointwise Constraints ◽  
The Magnetic Field

In order to generate a desired Kelvin (magnetic) force in a target subdomain moving along a prescribed trajectory, we propose a minimization problem with a tracking type cost functional. We use the so-called dipole approximation to realize the magnetic field, where the location and the direction of the magnetic sources are assumed to be fixed. The magnetic field intensity acts as the control and exhibits limiting pointwise constraints. We address two specific problems: the first one corresponds to a fixed final time whereas the second one deals with an unknown force to minimize the final time. We prove existence of solutions and deduce local uniqueness provided that a second-order sufficient condition is valid. We use the classical backward Euler scheme for time discretization. For both problems we prove the [Formula: see text]-weak convergence of this semi-discrete numerical scheme. This result is motivated by [Formula: see text]-convergence and does not require second-order sufficient condition. If the latter holds then we prove [Formula: see text]-strong local convergence. We report computational results to assess the performance of the numerical methods. As an application, we study the control of magnetic nanoparticles as those used in magnetic drug delivery, where the optimized Kelvin force is used to transport the drug to a desired location.

Crouzeix–Raviart Finite Element Approximation for the Parabolic Obstacle Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 273-292 ◽  
Author(s):  
Thirupathi Gudi ◽  
Papri Majumder
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Time Discretization ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Fully Discrete ◽  
Backward Euler ◽  
Space Discretization ◽  
Speed Of Propagation

AbstractWe introduce and study a fully discrete nonconforming finite element approximation for a parabolic variational inequality associated with a general obstacle problem. The method comprises of the Crouzeix–Raviart finite element method for space discretization and implicit backward Euler scheme for time discretization. We derive an error estimate of optimal order {\mathcal{O}(h+\Delta t)} in a certain energy norm defined precisely in the article. We only assume the realistic regularity {u_{t}\in L^{2}(0,T;L^{2}(\Omega))} and moreover the analysis is performed without any assumptions on the speed of propagation of the free boundary. We present a numerical experiment to illustrate the theoretical order of convergence derived in the article.

scholarly journals A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

Algorithms ◽  
2020 ◽  
Vol 13 (10) ◽  
pp. 242
Author(s):  
Gujji Murali Mohan Reddy ◽  
Alan B. Seitenfuss ◽  
Débora de Oliveira Medeiros ◽  
Luca Meacci ◽  
Milton Assunção ◽  
...  
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Piecewise Linear ◽  
Time Discretization ◽  
Comsol Multiphysics ◽  
Integral Term ◽  
Finite Element Software ◽  
Time Stepping ◽  
Backward Euler ◽  
Shaped Domain

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

Solution of two-dimensional vector wave equations via a mixed least squares method

2015 ◽  
Vol 32 (7) ◽  
pp. 1893-1907
Author(s):  
Maria Tchonkova
Keyword(s):  
Least Squares ◽  
Wave Equations ◽  
Least Squares Method ◽  
Equations Of Motion ◽  
Time Discretization ◽  
Weak Form ◽  
Time Step ◽  
Content Type ◽  
Vector Wave ◽  
Backward Euler

Purpose – The purpose of this paper is to present an original mixed least squares method for the numerical solution of vector wave equations. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. Findings – To test the performance of the method, it has been implemented in original computer codes, using object-oriented logic. One model problem has been solved: propagation of Rayleigh waves. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications.

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