Crank–Nicolson Leap-Frog Time Stepping Decoupled Scheme for the Fluid–Fluid Interaction Problems

2020 ◽  
Vol 84 (1) ◽  
Author(s):  
Lingzhi Qian ◽  
Xinlong Feng ◽  
Yinnian He
Keyword(s):  
Time Stepping ◽  
Fluid Interaction ◽  
Crank Nicolson

A linear, decoupled fractional time‐stepping method for the nonlinear fluid–fluid interaction

2019 ◽  
Vol 35 (5) ◽  
pp. 1873-1889 ◽  
Author(s):  
Jian Li ◽  
Pengzhan Huang ◽  
Chong Zhang ◽  
Gaihui Guo
Keyword(s):  
Time Stepping ◽  
Fluid Interaction ◽  
Nonlinear Fluid

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.

Decoupled Time Stepping Methods for Fluid-Fluid Interaction

2012 ◽  
Vol 50 (3) ◽  
pp. 1297-1319 ◽  
Author(s):  
Jeffrey M. Connors ◽  
Jason S. Howell ◽  
William J. Layton
Keyword(s):  
Time Stepping ◽  
Fluid Interaction

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.

Sign in / Sign up

Export Citation Format

Share Document