Convergence Analysis of High Order Algebraic Fractional Step Schemes for Time-Dependent Stokes Equations

2008 ◽  
Vol 46 (4) ◽  
pp. 1682-1703 ◽  
Author(s):  
Paola Gervasio
Keyword(s):  
Convergence Analysis ◽  
Stokes Equations ◽  
High Order ◽  
Time Dependent ◽  
Fractional Step ◽  
Order Algebraic

A stabilized fractional-step finite element method for the time-dependent Navier–Stokes equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yueqiang Shang ◽  
Qing Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Rate ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Fractional Step ◽  
Navier Stokes Equations ◽  
Subgrid Model ◽  
Element Method

Abstract We present a fractional-step finite element method based on a subgrid model for simulating the time-dependent incompressible Navier–Stokes equations. The method aims to the simulation of high Reynolds number flows and consists of two steps in which the nonlinearity and incompressibility are split into different steps. The first step of this method can be seen as a linearized Burger’s problem where a subgrid model based on an elliptic projection of the velocity into a lower-order finite element space is employed to stabilize the system, and the second step is a Stokes problem. Under mild regularity assumptions on the continuous solution, we obtain the stability of the numerical method, and derive error bound of the approximate velocity, which shows that first-order convergence rate in time and optimal convergence rate in space can be gotten by the method. Numerical experiments verify the theoretical predictions and demonstrate the promise of the proposed method, which show superiority of the proposed method to the compared method in the literature.

scholarly journals A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations

2015 ◽  
Vol 18 (4) ◽  
pp. 985-1011 ◽  
Author(s):  
Liang Pan ◽  
Kun Xu
Keyword(s):  
Stokes Equations ◽  
Kinetic Scheme ◽  
Second Order ◽  
High Order ◽  
Time Dependent ◽  
Navier Stokes ◽  
Third Order ◽  
A Cell ◽  
Flow Variables ◽  
Cell Interface

AbstractIn this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values, the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for high-order accuracy in the Discontinuous Galerkin method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. It can be faithfully used from the incompressible limit to the hypersonic flow computations, and many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique. Due to its multidimensional property of including both derivatives of flow variables in the normal and tangential directions of a cell interface, the viscous flow solution, especially those with vortex structure, can be accurately captured. With the same stencil of a second order scheme, numerical tests demonstrate that the current scheme is as robust as well-developed second-order shock capturing schemes, but provides more accurate numerical solutions than the second order counterparts.

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