Third-order-accurate semi-implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

2006 ◽  
Vol 51 (2) ◽  
pp. 221-233 ◽  
Author(s):  
Nikolay Nikitin
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations ◽  
Runge Kutta

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

2016 ◽  
Vol 20 (4) ◽  
pp. 1016-1044 ◽  
Author(s):  
Xiaodong Liu ◽  
Yidong Xia ◽  
Hong Luo ◽  
Lijun Xuan
Keyword(s):  
Comparative Study ◽  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations ◽  
Index 2 ◽  
Runge Kutta

AbstractA 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.

Advanced Numerical Methods for the Prediction of Tonal Noise in Turbomachinery—Part I: Implicit Runge–Kutta Schemes

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Graham Ashcroft ◽  
Christian Frey ◽  
Kathrin Heitkamp ◽  
Christian Weckmüller
Keyword(s):  
Stokes Equations ◽  
Temporal Integration ◽  
Aerodynamic Noise ◽  
Navier Stokes ◽  
Noise Generation ◽  
Academic Problems ◽  
Tonal Noise ◽  
Navier Stokes Equations ◽  
Runge Kutta ◽  
The Stability

This is the first part of a series of two papers on unsteady computational fluid dynamics (CFD) methods for the numerical simulation of aerodynamic noise generation and propagation. In this part, the stability, accuracy, and efficiency of implicit Runge–Kutta schemes for the temporal integration of the compressible Navier–Stokes equations are investigated in the context of a CFD code for turbomachinery applications. Using two model academic problems, the properties of two explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) schemes of second- and third-order accuracy are quantified and compared with more conventional second-order multistep methods. Finally, to assess the ESDIRK schemes in the context of an industrially relevant configuration, the schemes are applied to predict the tonal noise generation and transmission in a modern high bypass ratio fan stage and comparisons with the corresponding experimental data are provided.

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

2005 ◽  
Vol 21 (6) ◽  
pp. 542-549 ◽  
Author(s):  
Yuxin Ren ◽  
Yuxi Jiang ◽  
Miao'er Liu ◽  
Hanxin Zhang
Keyword(s):  
Stokes Equations ◽  
Projection Methods ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equations

scholarly journals Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations

2000 ◽  
Vol 35 (3) ◽  
pp. 177-219 ◽  
Author(s):  
Christopher A. Kennedy ◽  
Mark H. Carpenter ◽  
R.Michael Lewis
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Runge Kutta
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