HIGH-ORDER ACCURATE NUMERICAL SCHEMES FOR THE PARABOLIC EQUATION

2005 ◽  
Vol 13 (04) ◽  
pp. 613-639 ◽  
Author(s):  
EVANGELIA T. FLOURI ◽  
JOHN A. EKATERINARIS ◽  
NIKOLAOS A. KAMPANIS
Keyword(s):  
Parabolic Equation ◽  
Finite Difference ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
High Order ◽  
Curvilinear Coordinates ◽  
Underwater Sound ◽  
Numerical Schemes ◽  
Implicit Methods ◽  
Simple Boundary

Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

Aeroacoustic Predictions Using High-Order Shock-Capturing Schemes

2003 ◽  
Vol 2 (2) ◽  
pp. 175-192 ◽  
Author(s):  
John A. Ekaterinaris
Keyword(s):  
Numerical Solutions ◽  
Sound Propagation ◽  
Finite Difference Methods ◽  
High Order ◽  
Curvilinear Coordinates ◽  
Compressible Turbulence ◽  
Test Problems ◽  
Complex Flows ◽  
Time Marching ◽  
Implicit And Explicit

High-order accurate, finite-difference methods, such as the compact centered schemes with spectral-type or characteristic-based filters and the weighted essentially non-oscillatory (WENO) schemes, which are used in high resolution CFD solutions and for DNS or LES of compressible turbulence, are applied to aeroacoustics. Implicit and explicit schemes are used for time marching. The accuracy of the numerical solutions is evaluated for test problems. It is found that these methods are appropriate for sound propagation in complex flows that require use of curvilinear coordinates. Therefore they are applicable for the prediction of sound generation from both smooth subsonic flows, and transonic or supersonic flows with discontinuities.

A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

2021 ◽  
Author(s):  
Adrin Gharakhani
Keyword(s):  
Finite Difference ◽  
Poisson Equation ◽  
Finite Difference Methods ◽  
Neumann Boundary ◽  
Normal Derivative ◽  
High Order ◽  
Benchmark Problems ◽  
Flux Reconstruction ◽  
Reconstruction Method ◽  
Vorticity Transport

Abstract A compact high-order finite difference method on unstructured meshes is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the Flux Reconstruction Method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction to enforce the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference methods for the VTE, where the wall vorticity is approximated by finite differencing the second wall-normal derivative of the streamfunction, the new method applies a Neumann boundary condition for the diffusion of vorticity such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. This yields a wall vorticity with order of accuracy consistent with that of the overall solution. In this paper, the high-order VTE solver is formulated and results presented to demonstrate the accuracy and convergence rate of the Poisson solution, as well as the VTE solver using benchmark problems of 2-D flow in lid-driven cavity and backward facing step channel at various Reynolds numbers.

On the Solution of the Diffusion Equations by Numerical Methods

1966 ◽  
Vol 88 (4) ◽  
pp. 421-427 ◽  
Author(s):  
H. Z. Barakat ◽  
J. A. Clark
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Present Method ◽  
Finite Difference Methods ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Approximation Procedure ◽  
Implicit Methods ◽  
Difference Methods ◽  
Explicit Finite Difference

An explicit-finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, nonhomogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems are compared with the exact solution and with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in closer agreement with the exact solution than are those obtained by the other methods.

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