Computer Graphics
Curriculum Vitae

Image Registration Computer Vision Computat. Anatomy Point Clouds Surface Denoising Image Segmentation Medical Imaging Image Processing Compressible Fluids Incompressible Fluids Level Set Methods Vortex Sheets



    Hyperbolic partial differential equations describe a wide range of wave-propagation and transport phenomena arising in nearly every scientific and engineering discipline.  Here are a few animations of time-dependent solutions to 1D Euler Equations, which represent the behavior of a compressible gas.  These movies illustrate that third generation Essentially Non-Oscillatory (ENO) methods give much more accurate approximations than first generation schemes, such as Lax-Friedrichs, do. ENO schemes, originally developed by Harten, Engquist, Osher, and Chakravarthy, capture shocks, rarefactions, and contact discontinuities accurately.



Euler Equations

Methods \ Test Cases Test Case #1 Test Case #2
Lax-Friedrichs view view
McCormack view view
Steger-Warming VFS view view
ENO(2)-SW-VFS-RK3 view view
WENO(3)-SW-VFS-RK3 view view
ENO(3)-SW-VFS-RK3 view view
WENO(5)-SW-VFS-RK3 view view
ENO(5)-SW-VFS-RK3 view view
WENO(9)-SW-VFS-RK3 view view

Analyze results in .jpg format


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Oblique Shock Problem


   We consider the problem of an oblique shock, generated by a supersonic flow over a sharp wedge.  The problem involves the grid generation.  The Euler equations are solved in two spatial dimensions on nonrectangular grid by employing the finite volume formulations.  The animations below illustrate two different approaches.  The three shocks and four regions of homogeneous pressure are found.



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Igor Yanovsky