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Results tagged with finite-differences
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user 11260
2
votes
Algorithmically finding mixed-derivative coefficients from finite differences
An algorithm for this purpose has been developed by B. Fornberg in Calculation of Weights in Finite Difference Formulas.
It has been implemented in Mathematica, see the documentation:
- 188k
2
votes
Accepted
Numerically calculating the divergence of a set of oriented points
Interpolation of vector fields (a notoriously tricky process) to determine the divergence can be avoided by using a line integral definition of the divergence via Stokes' theorem. You can find a discu …
- 188k
1
vote
Laplace equation, medium discontinuity and finite difference method
A surface charge will accumulate on the interface where the dielectric constant has a discontinuity. You need to calculate this surface charge and include it into the discretised Poisson equation. Her …
- 188k
2
votes
High order difference operator applied to 1/u
You can apply the formula
$$\Delta^p f(x)=\sum_{k=0}^p{p\choose k}(-1)^{p-k}f(x+k)$$
to $f(x)=1/u(x)$. Further simplification will need knowledge how $u(x)$ depends on $x$.
For example, if $u(x)=x$ on …
- 188k
5
votes
Accepted
Change of variable formulas in discrete calculus?
In terms of the differential operator $\partial_x\equiv d/dx$ one has $f(x+h)=e^{h\partial_x}f(x)$, hence
$$\Delta_h=e^{h\partial_x}-1.$$
Upon Fourier transformation, $\hat{f}(k)=\int_{-\infty}^\infty …
- 188k