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I have the following doubts regarding the Lax-Wendroff scheme for nonlinear conservation laws.

For a scalar conservation law given by $u_t+f(u)_x=0,$ where $f$ is not linear?.

Lax-Wendroff scheme is conservative and consistent scheme but not monotone. How to show that it converges to the weak solution? Standard proof requires monotone conservative and consistency... But monotonicity is not true in this case.

If not, is it possible to construct an initial data and show that Lax-Wendroff scheme does not converge to the weak solution

Rosy
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    Thanks for the comment....Yes it's not monotone.. So how to prove the convergence? – Rosy Apr 06 '20 at 18:57
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    You can prove convergence towards smooth (strong) solutions in L2 norm, with second-order accuracy in space and time. However, I don't know a proof of weak convergence. I even doubt that this property is true. (LW is neither monotonous nor TVD) – EditPiAf Apr 13 '20 at 11:17
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    @EditPiAf , I was trying to construct a sequence and show that tv blows up but could not succeed. When i computed the solution for Riemann data admitting shocks, I noticed hump near the discontinuity, whose area was decreasing with decreasing the mesh size.. Let me know your thoughts on this. – Rosy May 06 '20 at 19:22
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    You may be able to go further in the linear case $f(u) = au$ where the TV may also be increasing, see e.g. this post – EditPiAf May 06 '20 at 20:58

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