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.


1 Answer 1


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .