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A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations

Part of: Basic methods in fluid mechanics Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 October 2016

Xiaodong Liu ,
Yidong Xia ,
Hong Luo  and
Lijun Xuan
Xiaodong Liu*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA
Yidong Xia*
Affiliation:
Department of Energy Resource Recovery & Sustainability, Idaho National Laboratory, Idaho Falls, ID 83415, USA
Hong Luo*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA
Lijun Xuan*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA
*
*Corresponding author. Email addresses:xliu29@ncsu.edu (X. Liu), yidongxia@gmail.com (Y. Xia), hong_luo@ncsu.edu (H. Luo), lxuan@ncsu.edu (L. Xuan)
*Corresponding author. Email addresses:xliu29@ncsu.edu (X. Liu), yidongxia@gmail.com (Y. Xia), hong_luo@ncsu.edu (H. Luo), lxuan@ncsu.edu (L. Xuan)
*Corresponding author. Email addresses:xliu29@ncsu.edu (X. Liu), yidongxia@gmail.com (Y. Xia), hong_luo@ncsu.edu (H. Luo), lxuan@ncsu.edu (L. Xuan)
*Corresponding author. Email addresses:xliu29@ncsu.edu (X. Liu), yidongxia@gmail.com (Y. Xia), hong_luo@ncsu.edu (H. Luo), lxuan@ncsu.edu (L. Xuan)
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Abstract

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

Keywords

Implicit time integrationRosenbrock-Wannerdiscontinuous GalerkinWENONavier-Stokes

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76M10: Finite element methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 4 , October 2016 , pp. 1016 - 1044
Copyright
Copyright © Global-Science Press 2016 

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