scholarly journals Intrusive generalized polynomial chaos with asynchronous time integration for the solution of the unsteady Navier-Stokes equations

2021 ◽  
pp. 104952
Author(s):  
P. Bonnaire ◽  
P. Pettersson ◽  
C.F. Silva
Keyword(s):  
Polynomial Chaos ◽  
Stokes Equations ◽  
Time Integration ◽  
Navier Stokes ◽  
Generalized Polynomial Chaos ◽  
Navier Stokes Equations ◽  
Generalized Polynomial

Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos

2001 ◽  
Vol 124 (1) ◽  
pp. 51-59 ◽  
Author(s):  
Dongbin Xiu ◽  
Didier Lucor ◽  
C.-H. Su ◽  
George Em Karniadakis
Keyword(s):  
Flow Structure ◽  
Polynomial Chaos ◽  
Stokes Equations ◽  
Hermite Polynomials ◽  
Weak Form ◽  
Galerkin Projection ◽  
Navier Stokes ◽  
Generalized Polynomial Chaos ◽  
Navier Stokes Equations ◽  
Generalized Polynomial

We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flow-structure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener (1938) and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to second-order oscillators to demonstrate convergence, and subsequently is coupled to incompressible Navier-Stokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortex-induced vibrations.

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.

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