Analysis of Stabilized Crank-Nicolson Time-Stepping Scheme for the Evolutionary Peterlin Viscoelastic Model

2020 ◽  
Vol 41 (13) ◽  
pp. 1611-1641
Author(s):  
S. S. Ravindran
Keyword(s):  
Viscoelastic Model ◽  
Time Stepping ◽  
Crank Nicolson

scholarly journals On the Convergence of a Crank–Nicolson Fitted Finite Volume Method for Pricing American Bond Options

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Xiaoting Gan ◽  
Dengguo Xu
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Nonlinear Partial Differential Equation ◽  
System Of Nonlinear Equations ◽  
Partial Differential ◽  
Time Stepping ◽  
Bond Options ◽  
Volume Method ◽  
Coupon Bond ◽  
Crank Nicolson

This paper develops and analyses a Crank–Nicolson fitted finite volume method to price American options on a zero-coupon bond under the Cox–Ingersoll–Ross (CIR) model governed by a partial differential complementarity problem (PDCP). Based on a penalty approach, the PDCP results in a nonlinear partial differential equation (PDE). We then apply a fitted finite volume method for the spatial discretization along with a Crank–Nicolson time-stepping scheme for the PDE, which results in a nonlinear algebraic equation. We show that this scheme is consistent, stable, and monotone, and hence, the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. To solve the system of nonlinear equations effectively, an iterative algorithm is established and its convergence is proved. Numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the new numerical method.

Adaptive Multigrid Methods for Extended Fluid-Structure Interaction (eXFSI) Problem: Part I — Mathematical Modelling

2015 ◽  
Author(s):  
Bhuiyan Shameem Mahmood Ebna Hai ◽  
Markus Bause
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Variational Formulation ◽  
Fluid Structure Interaction ◽  
Multigrid Methods ◽  
Time Step ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Time Stepping ◽  
Crank Nicolson

This contribution is the first part of three papers on Adaptive Multigrid Methods for eXtended Fluid-Structure Interaction (eXFSI) Problem, where we introduce a monolithic variational formulation and solution techniques. In a monolithic nonlinear fluid-structure interaction (FSI), the fluid and structure models are formulated in different coordinate systems. This makes the FSI setup of a common variational description difficult and challenging. This article presents the state-of-the-art of recent developments in the finite element approximation of FSI problem based on monolithic variational formulation in the well-established arbitrary Lagrangian Eulerian (ALE) framework. This research will focus on the newly developed mathematical model of a new FSI problem which is called eXtended Fluid-Structure Interaction (eXFSI) problem in ALE framework. This model is used to design an on-live Structural Health Monitoring (SHM) system in order to determine the wave propagation in moving domains and optimum locations for SHM sensors. eXFSI is strongly coupled problem of typical FSI with a wave propagation problem on the fluid-structure interface, where wave propagation problems automatically adopted the boundary conditions from of the typical FSI problem at each time step. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of eXFSI problem, the analysis of various fluid-mesh motion techniques and comparison of different second-order time-stepping schemes. This work consists of the investigation of different time stepping scheme formulations for a nonlinear FSI problem coupling the acoustic/elastic wave propagation on the fluid-structure interface. Temporal discretization is based on finite differences and is formulated as an one step-θ scheme; from which we can consider the following particular cases: the implicit Euler, Crank-Nicolson, shifted Crank-Nicolson and the Fractional-Step-θ schemes. The nonlinear problem is solved with Newton’s method whereas the spatial discretization is done with a Galerkin finite element scheme. To control computational costs we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaptation during the computation. The implementation is accomplished via the software library package DOpElib and deal.II for the computation of different eXFSI configurations.

scholarly journals An Explicit Stabilized Runge–Kutta–Legendre Super Time-Stepping Scheme for the Richards Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ramesh Chandra Timsina ◽  
Harihar Khanal ◽  
Kedar Nath Uprety
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Richards Equation ◽  
One Dimensional ◽  
Time Stepping ◽  
Runge Kutta ◽  
Crank Nicolson

We solve one-dimensional Kirchhof transformed Richards equation numerically using finite difference method with various time-stepping schemes, forward in time central in space (FTCS), backward in time central in space (BTCS), Crank–Nicolson (CN), and a stabilized Runge–Kutta–Legendre super time-stepping (RKL), and compare their performances.

scholarly journals A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system

2017 ◽  
Vol 8 (1) ◽  
pp. 43-62
Author(s):  
S. S. Ravindran
Keyword(s):  
Heat Equation ◽  
Mixed Finite Element ◽  
Mhd Equations ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping ◽  
Crank Nicolson

Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics of electro-magnetically and thermally driven flows,involving MHD equations coupled with heat equation. We introduce a partitioned method that allows one to decouplethe MHD equations from the heat equation at each time step and solve them separately. The extrapolated Crank-Nicolson time-stepping scheme is used for time discretizationwhile mixed finite element method is used for spatial discretization. We derive optimal order error estimates in suitable norms without assuming any stability condition or restrictions on the time step size. We prove the unconditional stability of the scheme. Numerical experiments are used to illustrate the theoretical results.

Adaptive Finite Elements Simulation Methods and Applications for Monolithic Fluid-Structure Interaction (FSI) Problem

2014 ◽  
Author(s):  
Bhuiyan Shameem Mahmood Ebna Hai ◽  
Markus Bause
Keyword(s):  
Fluid Dynamics ◽  
Structural Analysis ◽  
Solid Mechanics ◽  
Stokes Equations ◽  
Fluid Structure Interaction ◽  
Finite Difference Schemes ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Time Stepping ◽  
Crank Nicolson

Will an aircraft wing have the structural integrity to withstand the forces or fail when it’s racing at a full speed? Fluid-structure interaction (FSI) analysis can help you to answer this question without the need to create costly prototypes. However, combining fluid dynamics with structural analysis traditionally poses a formidable challenge for even the most advanced numerical techniques due to the disconnected, domain-specific nature of analysis tools. In this paper, we present the state-of-the-art in computational FSI methods and techniques that go beyond the fundamentals of computational fluid and solid mechanics. In fact, the fundamental rule require transferring results from the computational fluid dynamics (CFD) analysis as input into the structural analysis and thus can be time-consuming, tedious and error-prone. This work consists of the investigation of different time stepping scheme formulations for a nonlinear fluid-structure interaction problem coupling the incompressible Navier-Stokes equations with a hyperelastic solid based on the well established Arbitrary Lagrangian Eulerian (ALE) framework. Temporal discretization is based on finite differences and a formulation as one step-θ scheme, from which we can extract the implicit euler, crank-nicolson, shifted crank-nicolson and the fractional-step-θ schemes. The ALE approach provides a simple, but powerful procedure to couple fluid flows with solid deformations by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. The goal of this work is the development of concepts for the efficient numerical solution of FSI problem and the analysis of various fluid-mesh motion techniques, a comparison of different second-order time-stepping schemes. The time discretization is based on finite difference schemes whereas the spatial discretization is done with a Galerkin finite element scheme. The nonlinear problem is solved with Newton’s method. To control computational costs, we apply a simplified version of a posteriori error estimation using the dual weighted residual (DWR) method. This method is used for the mesh adaption during the computation. The implementation using the software library package DOpElib and deal.II serves for the computation of different fluid-structure configurations.

IMPROVED ACCURACY FOR LOCALLY ONE-DIMENSIONAL METHODS FOR PARABOLIC EQUATIONS

2001 ◽  
Vol 11 (09) ◽  
pp. 1563-1579 ◽  
Author(s):  
JIM DOUGLAS ◽  
SEONGJAI KIM
Keyword(s):  
Parabolic Equations ◽  
Wave Equations ◽  
Computational Cost ◽  
Implicit Time ◽  
Differential Problem ◽  
Time Stepping ◽  
Backward Euler ◽  
The Cost ◽  
Improved Accuracy ◽  
Crank Nicolson

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank–Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.

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