This topic is a copy of that one on Mathematica.StackExchange. The question is about automatical derivation of coefficients of numerical scheme on a variable stencil. So, lets consider linear 1d transport equation

$$(1)\qquad u_t+u_x=0.$$

To discretize (1), let's consider the following stencil:

$$(2)\qquad (t^{n+1},x_{j-1}),\,(t^{n},x_{j-1}),\,(t^{n},x_{j}).$$

Then, let's consider the following approximation of (1) on (2):

$$(3)\qquad u_j^{n+1} = a_1u_{j-1}^n+a_2u_j^n,$$

where $a_1,\,a_2$ to be determined by Taylor series expansion. My first step seems to be as follows:

eqn = D[u[x, t], t] + D[u[x, t], x] == 0
approx = Series[
u[x, t + \[Tau]], {t, 0, 2}] - (a1 Series[u[x - h, t], {x, 0, 2}] + a2 Series[u[x, t], {x, 0, 2}])

The question is how to derive differential consequence from (1) to obtain system of two equation on $a_i,\,i=1,2$? Means, that we hold stencil (2) and approximation (3), consequently. The question is how to obtain $a_1,a_2$ by unknow coefficient method with the usage of differential

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