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Suppose $F(f,f_t,f_x,...)=0$ is a PDE of degree $d>0$ subject to some I.C. such that there exists a unique solution $f$ to it. Then under what conditions does a finite difference method converge (if there are any) to the true solution?

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    $\begingroup$ This question is much too broad to answer in general. Typically, if a scheme is consistent and stable then it is convergent (the Lax equivalence theorem asserts this for well posed linear PDE, for example) but even this will fail for some relatively simple PDE. $\endgroup$ – User8128 Sep 3 '17 at 16:11
  • $\begingroup$ Additionally convergence can only be proven in theory. When you solve the PDE on a computer things might be different as well. $\endgroup$ – P. Siehr Sep 4 '17 at 8:32
  • $\begingroup$ Why, what so you mean? $\endgroup$ – AIM_BLB Sep 4 '17 at 13:24
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    $\begingroup$ Is it a linear pde? As far as I know, there has never been, and likely never will be, a useful general result for when a finite difference scheme will converge for any nonlinear pde. $\endgroup$ – Merkh Sep 6 '17 at 16:47
  • $\begingroup$ You may have a look to this post, which illustrates that stability considerations are highly problem- and method-dependent. $\endgroup$ – Harry49 Sep 17 '17 at 14:06

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