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# Resolution with a non constant v velocity

Now, the v velocity is not a constant and the 2-D Burgers equation possesses two nolinear terms. We propose here to calculate this speed with the continuity equation defined by

 (2.4)
So the new problem to solve has no physical representation. In fact, if we consider a physical problem describes by the 2-D Burgers equation issue to the simplification of the Navier-Stokes equations, it signified that the v velocity does not depends of x, y and z variables. The introduction of the continuity equation imposed the dependence of these variables for v velocity. Physically, it is impossible to consider the continuity equation and the Burgers' equation together.
However, it may be interresting to take a non constant v velocity in order to see a not uniform transversal convection.

Equation 2.4 gives

 (2.5)
with v=v(x,y,t)

We can see that the v velocity is all more so great since the u velocity gradients are important. This is true where the shock is because it is defined by its discontinuity, so its velocity gradients. With this v velocity we can expect to see the highest transversal convection where the shock is.

Numerically, we approximate the term  by a central difference defined by

And v can be approximated by a trapezoidal method defined by

To solve the problem, we must initialized the calculation (see appendix Appendix B) by an arbitrary v velocity : we choose.

Next:The numerical problem Up:The physical problem Previous:Resolution with a constant   Contents

Alban Depoutre
2000-11-21