I am analyzing a quasi one dimensional supersonic flow field using Method of Characteristics.The medium is air. I am implementing Method of Characteristics numerically in the regions of continuous flow and jump conditions at discontinuities. Time step used is 7.5e-06 sec. During the simulation, I find that the solution becomes highly oscillatory and blows up in the regions adjoining contact discontinuity.
What kind of discontinuity it is? Before implementing numerical schemes, you should try to make sure that the equation formed is continuous in the given limits. Although, it is not possible to check in all cases, most discontinuities are removable. Explore the different methods to achieve that. You can try changing the variable through proper substitution so that the integrating variable is continuous in the new range of integration. Also, check if problem exists if you vary the time-step by different orders of magnitude. Besides, check numerical precision as well and your desired tolerance (accuracy) in the results.
Also, sometimes if the same equation is reformulated in some other form, it makes numerical implementation smooth. One of the important factor behind oscillation of numerical results is round-off errors due to subtraction cancellation, which often goes un-noticed. Your time step used is 7.5e-06, which is small and could lead to such numerical discrepancies. Also, make sure you have declared your variables in double precision. I would suggest you to go through section 5 of my forthcoming article. You will get an idea there. The link is:
https://www.researchgate.net/publication/280739383_Spatial_dispersion_of_elastic_waves_in_a_bar_characterized_by_tempered_nonlocal_elasticity
Technical Report Spatial dispersion of elastic waves in a bar characterized b...
In contact discontinuities, the only field that has a jump is density. Pressure and velocity are constant across the discontinuity. Many numerical methods have problems at contact discontinuities, but usually what happens is that the discontinuity is smoothed out by numerical diffusion, so what you are observing is unusual. You may find some explanations in the books
LeVeque, Randall J. Finite volume methods for hyperbolic problems. Vol. 31. Cambridge university press, 2002.
Toro, Eleuterio F. Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Science & Business Media, 2009.
Best regards
Luca Bonaventura
Thanks Vikash. I am presently looking at some methods to smooth out the discontinuity gradually over many time steps. I have tried out bigger time step. But the scheme becomes highly unstable. I will look at other aspects as suggested by you.
Instability in larger time steps? Not a good sign. There is possibly some serious misformulation or code implementation.
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