Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

2011 ◽  
Vol 59 (2) ◽  
pp. 271-300 ◽  
Author(s):  
Zhiyong Si
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Method Of Characteristics ◽  
Second Order ◽  
Time Dependent ◽  
Navier Stokes ◽  
Defect Correction ◽  
Modified Method ◽  
Stokes Problems ◽  
Modified Method Of Characteristics

scholarly journals MODIFIED METHOD OF CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME DEPENDENT NAVIER-STOKES PROBLEMS

2015 ◽  
Vol 20 (5) ◽  
pp. 658-680 ◽  
Author(s):  
Zhiyong Si ◽  
Yunxia Wang ◽  
Xinlong Feng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Method Of Characteristics ◽  
Numerical Results ◽  
Time Dependent ◽  
Navier Stokes ◽  
Variational Multiscale ◽  
Modified Method ◽  
Modified Method Of Characteristics ◽  
Element Method

In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent NavierStokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

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