Boundary update via resolvent for fluid-structure interaction

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Martina Bukač ◽  
Catalin Trenchea
Keyword(s):  
Quadrature Rule ◽  
Second Order ◽  
Energy Estimates ◽  
Euler Scheme ◽  
Discrete Problem ◽  
Unconditionally Stable ◽  
Backward Euler ◽  
Second Order Convergence ◽  
The Stability ◽  
Forward Euler

AbstractWe propose a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous, incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler - Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first solved using a Backward Euler scheme, (ii) the velocities of fluid and structure are updated on the boundary via a second-order consistent resolvent operator, and then (iii) the structure and fluid sub-problems are solved again, using a Forward Euler scheme. The stability analysis based on energy estimates shows that the scheme is unconditionally stable. Error analysis of the semi-discrete problem yields second-order convergence in time. The two numerical examples confirm theoretical convergence analysis results and show an excellent agreement between the proposed partitioned scheme and the monolithic scheme.

scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

Uniform Second-Order Hybrid Schemes on Bakhvalov-Shishkin Mesh for Quasi-Linear Convection-Diffusion Problems

2013 ◽  
Vol 871 ◽  
pp. 135-140 ◽  
Author(s):  
Quan Zheng ◽  
Xue Zheng Li ◽  
Yu Feng Liu
Keyword(s):  
Shishkin Mesh ◽  
Second Order ◽  
Order Estimate ◽  
Central Difference ◽  
Hybrid Schemes ◽  
Central Difference Scheme ◽  
Convection Diffusion ◽  
Second Order Convergence ◽  
The Stability ◽  
Diffusion Problems

In this paper, we propose a class of hybrid difference schemes combining the central difference scheme and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh for solving quasi-linear singularly perturbed convection-diffusion boundary value problems. Point-wise second-order convergence uniform in the perturbation is proved clearly by using the-stability. The numerical experiments support the schemes and the uniform second-order estimate.

An Unconditionally Stable and High-Order Convergent Difference Scheme for Stokes’ First Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

2017 ◽  
Vol 10 (3) ◽  
pp. 597-613 ◽  
Author(s):  
Cuicui Ji ◽  
Zhizhong Sun
Keyword(s):  
Fractional Derivative ◽  
Difference Scheme ◽  
Second Order ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Unconditionally Stable ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid ◽  
Second Order Convergence ◽  
Convergent Difference Scheme

AbstractThis article is intended to fill in the blank of the numerical schemes with second-order convergence accuracy in time for nonlinear Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. A linearized difference scheme is proposed. The time fractional-order derivative is discretized by second-order shifted and weighted Gr¨unwald-Letnikov difference operator. The convergence accuracy in space is improved by performing the average operator. The presented numerical method is unconditionally stable with the global convergence order of in maximum norm, where τ and h are the step sizes in time and space, respectively. Finally, numerical examples are carried out to verify the theoretical results, showing that our scheme is efficient indeed.

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

2017 ◽  
Vol 6 (3) ◽  
Author(s):  
D. Shakti ◽  
J. Mohapatra
Keyword(s):  
Adaptive Grid ◽  
Richardson Extrapolation ◽  
Euler Scheme ◽  
Backward Euler Scheme ◽  
Monitor Function ◽  
Backward Euler ◽  
The Stability ◽  
Perturbation Problems ◽  
Equidistribution Principle ◽  
Richardson Extrapolation Technique

AbstractA nonlinear singularly perturbed boundary value problem depending on a parameter is considered. First, we solve the problem using the backward Euler finite difference scheme on an adaptive grid. The adaptive grid is a special nonuniform mesh generated through equidistribution principle by a positive monitor function depending on the solution. The behavior of the solution, the stability and the error estimates are discussed. Then, the Richardson extrapolation technique is applied to improve the accuracy of the computed solution associated to the backward Euler scheme. The proofs of the uniform convergence for the backward Euler scheme and the Richardson extrapolation are carried out. Numerical experiments validate the theoretical estimates and indicates that the estimates are sharp.

Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems

2017 ◽  
Vol 17 (2) ◽  
pp. 323-335
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Level Scheme ◽  
Order Approximation ◽  
Asymptotic Properties ◽  
Adjoint Operator ◽  
Second Order ◽  
Evolutionary Equation ◽  
Euler Scheme ◽  
Backward Euler Scheme ◽  
First Order ◽  
Backward Euler

AbstractSchemes with the second-order approximation in time are considered for numerically solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Padé polynomial approximation is unconditionally stable. It demonstrates good asymptotic properties in time and provides an adequate evolution in time for individual harmonics of the solution (has spectral mimetic (SM) stability). In fact, the only drawback of this scheme is the necessity to solve an equation with an operator polynomial of second degree at each time level. We consider modifications of these schemes, which are based on solving equations with operator polynomials of first degree. Such computational implementations occur, for example, if we apply the fully implicit two-level scheme (the backward Euler scheme). A three-level modification of the SM-stable scheme is proposed. Its unconditional stability is established in the corresponding norms. The emphasis is on the scheme, where the numerical algorithm involves two stages, namely, the backward Euler scheme of first order at the first (prediction) stage and the following correction of the approximate solution using a factorized operator. The SM-stability is established for the proposed scheme. To illustrate the theoretical results of the work, a model problem is solved numerically.

Stability of radical anions derived from substituted 5-nitrofurans investigated by ESR spectroscopy

1985 ◽  
Vol 50 (7) ◽  
pp. 1594-1601 ◽  
Author(s):  
Jiří Klíma ◽  
Larisa Baumane ◽  
Janis Stradinš ◽  
Jiří Volke ◽  
Romualds Gavars
Keyword(s):  
Radical Anion ◽  
High Stability ◽  
Esr Spectroscopy ◽  
Reaction Path ◽  
Order Reaction ◽  
Second Order ◽  
Radical Anions ◽  
Second Order Reaction ◽  
Unpaired Electron Density ◽  
The Stability

It has been found that the decay in dimethylformamide and dimethylformamide-water mixtures of radical anions in five of the investigated 5-nitrofurans is governed by a second-order reaction. Only the decay of the radical anion generated from 5-nitro-2-furfural III may be described by an equation including parallel first- and second-order reactions; this behaviour is evidently caused by the relatively high stability of the corresponding dianion, this being an intermediate in the reaction path. The presence of a larger conjugated system in the substituent in position 2 results in a decrease of the unpaired electron density in the nitro group and, consequently, an increase in the stability of the corresponding radical anions.

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 Stability of a one-dimensional morphoelastic model for post-burn contraction

2021 ◽  
Vol 83 (3) ◽  
Author(s):  
Ginger Egberts ◽  
Fred Vermolen ◽  
Paul van Zuijlen
Keyword(s):  
Wound Healing ◽  
Residual Stresses ◽  
Truncation Error ◽  
Signaling Molecules ◽  
Discrete Problem ◽  
One Dimensional ◽  
Stability Constraint ◽  
Biological Interpretation ◽  
The Stability ◽  
The One

AbstractTo deal with permanent deformations and residual stresses, we consider a morphoelastic model for the scar formation as the result of wound healing after a skin trauma. Next to the mechanical components such as strain and displacements, the model accounts for biological constituents such as the concentration of signaling molecules, the cellular densities of fibroblasts and myofibroblasts, and the density of collagen. Here we present stability constraints for the one-dimensional counterpart of this morphoelastic model, for both the continuous and (semi-) discrete problem. We show that the truncation error between these eigenvalues associated with the continuous and semi-discrete problem is of order $${{\mathcal {O}}}(h^2)$$ O ( h 2 ) . Next we perform numerical validation to these constraints and provide a biological interpretation of the (in)stability. For the mechanical part of the model, the results show the components reach equilibria in a (non) monotonic way, depending on the value of the viscosity. The results show that the parameters of the chemical part of the model need to meet the stability constraint, depending on the decay rate of the signaling molecules, to avoid unrealistic results.

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