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Yes, such rules can be derived in the same way that is done for polynomials. Given arbitrary basis functions $\phi_i(x)$, the quadrature rule $$\int f(x) dx \approx \sum_i c_i f(x_i)$$ will be exact …

Any analysis of global error must include information about how local errors are amplified in subsequent steps. So your statement I know that the local error at each step of Euler's method is …

I believe the intended reference regarding parabolic PDEs is: Fritz, John. On integration of parabolic equations by difference methods: I. Linear and quasi-linear equations for the infinite inter …

As @Oleg Eroshkin has already pointed out in the comments, this is closely related to solving a linear algebraic system. You could take absolute values and then logs to obtain a linear system of the …

Very many numerical methods for partial differential equations compute derivatives from values on something that is not a regular grid (they use unstructured grids). Usually these are finite element …

In general, when solving a PDE numerically, both the spatial and the temporal discretization will (of course) contribute to the local truncation error and hence to the global error. In many equations …

Obviously, within the realm of piecewise-smooth functions one can find examples where any derivative-based approach fails. I believe you're looking for the term "derivative free optimization". He …

There is a whole subfield of applied mathematics devoted to developing ODE solvers and understanding their properties. Consequently, there are thousands of relevant papers, and not much more can be s …

This is not a research-level question; the answers can be found in any undergraduate text on the subject. it looks to me as if f(y,t) is nonlinear the system of equations determining ki will also …

There are several good answers already, but none of them so far deal with high performance computing, which nowadays means computing on tens or hundreds of thousands of cores. I am aware of only thre …

You could use the method of lines to solve this PDE. If you use an explicit finite difference method, you will need to take a rather small time step (${\mathcal O(\Delta x^3))}$ due to the $u_{xxx}$ …

I faced this problem a few years ago. In that case, I obtained a satisfactory approach along the lines of one of your suggestions. Specifically, I found that the eigenvectors changed relatively slow …

A very straightforward explanation is given in R.J. LeVeque's text. In Chapter 2 there are simple explanations of how to show convergence for the linear problem in both the maximum norm and the Eucli …

Regarding 1 and 2: Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in provably (pointwise) convergent schemes. At least for traditional method …

For implicit methods, you can achieve order $2s$ with $s$ stages. Note that this result is the same if one considers the simpler problem of numerical integration (quadrature). Update as of 2024: a 16 …