A fast second‐order difference scheme for the space–time fractional equation

2019 ◽  
Vol 35 (4) ◽  
pp. 1326-1342 ◽  
Author(s):  
Weiyan Xu ◽  
Hong Sun
Keyword(s):  
Difference Scheme ◽  
Space Time ◽  
Second Order ◽  
Order Difference ◽  
Fractional Equation

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
J. Granata ◽  
L. Xu ◽  
Z. Rusak ◽  
S. Wang
Keyword(s):  
Numerical Simulation ◽  
Reynolds Number ◽  
Difference Scheme ◽  
Second Order ◽  
Swirling Flows ◽  
Inviscid Limit ◽  
Simulation Algorithm ◽  
Spatial Integration ◽  
Order Difference ◽  
Breakdown Process

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

Stability-Controllable Second-order Difference scheme for convection term

1998 ◽  
Vol 7 (2) ◽  
pp. 119-130 ◽  
Author(s):  
Ming-Jiu Ni ◽  
Wen-Quan Tao ◽  
Shang-Jin Wang
Keyword(s):  
Difference Scheme ◽  
Second Order ◽  
Convection Term ◽  
Order Difference
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