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Suppose you want to write a finite difference scheme for a linear PDE of the form

$ V_t + x V_x + x^2 V_{xx} =0 $,

where the subscripts x,t denote derivatives with respect to space and time respectively. I know that using Taylor expansions, one can derive forward/backward or theta schemes for $V_t$, $V_x$ and $V_{xx}$. However, how do I deal with the terms $x V_x$ and $x^2 V_{xx}$ ? Is it okay (in terms of convergence and stability) to e.g. approximate $x V_x(t,x_0)$ by something like $x_0 (\frac{V(t,x_1)-V(t,x_0)}{x_1-x_0})$?

I've searched the web but I can't find any useful links. Even some useful keywords would be very much appreciated.

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  • $\begingroup$ This is a linear PDE. $\endgroup$ – Artem Jan 18 '14 at 0:00
  • $\begingroup$ Thanks. Obviously I thought that (non)linear refers to the coefficients of the PDE, but it refers to the dependent variable V and its derivatives only. $\endgroup$ – Stochast Jan 18 '14 at 10:28
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I hope you find some good answers to this question - I have not been able to. I can point you to a few books where this is (only briefly) mentioned:

  • Randall, Leveque. Finite Difference Methods: Section 2.15
  • Granville, Sewell. Numerical Solutions of Ordinary and Partial Differential Equations: Section 4.2
  • Knabner, Angerman. Numerical Methods for Elliptic and Parabolic PDE: Section 1.2
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