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As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$u_x$' of a desired solution '$u$' at the point $x_n$ in the domain is just a linear combination $\displaystyle \sum_{i=0}^Nc_i u_i,$ where, $u_i=u(x_i)$.

That is, $\displaystyle (u_x)_n=u_x(x_n)\approx c_0u_0+c_1u_1+...+c_Nu_N$.

For instance, if $c_i=0,$ for $i=0,1,2,...,n-1,n+2,...,N$, $c_n=\frac{-1}{h}$, and $c_{n+1}=\frac{1}{h},$ with $h=x_{n+1}-x_n$ for all $n$ (equal sub interval length), then the approximation is said to be a 'forward difference approximation'. In a similar way, we can have backward differences, central differences, finite differences on non-uniform grids, non standard finite differences (NSFD), etc.

However, while analyzing the differential quadrature approximation (DQM),

\begin{equation}\label{DQM} u_{x}(x_{i})=\left.\dfrac{du}{dx}\right\vert_{x_{i}} \approx \sum_{j=1}^{N} a_{ij} u(x_{j}),\textrm{for}~ i=1,2,\dots,N, \end{equation}

I feel that none of the authors introduce it as a finite difference method.

Doubt: What exactly is the conceptual difference of DQM over FDM?,

Whether it is due to the number of non-zero coefficients $c_i$'s to be considered? OR

Due to the difference in methods of finding the coefficients $c_i$? We bother about the coefficients from the beginning itself while applying FDMs, whereas we need some test functions for DQM.

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1 Answer 1

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The linear “integral” operator $L$ applied to a function $u$ can be written as

$$L u(x_i) \approx \sum_{j} a_{ij} u(x_j) $$

where the coefficients are determined by many ways, usually. You just treat differential operator as some kind of integral operator. The original work by Bellman seemed to use a linear system.

$$L \phi(x_i) = \sum_{j} a_{ij} \phi(x_j) $$

where $\phi$ are taken through polynomials of various degrees, for instance, the Lagrange polynomials on the nodes. Then you will obtain a set of coefficients for DQM.

If you take $\phi$ as "hat function" (its integrals) at each node, then you will get the usual finite difference method.

If you take Fourier basis for $\phi$, then you will get the pseudo spectral method.

That is, the difference is simply what basis functions are you applying.

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