# Tag Info

### What is the advantage of using a particular RK Scheme?

You're looking only at the errors themselves and not other properties of the solution. There are sometimes good cases to consider lower-order schemes because they better preserve important ...
• 10.4k
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### Why can ill-conditioned linear systems be solved precisely?

Added after my initial answer: It appears to me now that the author of the referenced paper is giving condition numbers (apparently 2-norm condition numbers but possibly infinity-norm condition ...
• 18.9k
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### How to compute all the eigenvalues of a large sparse matrix using matlab?

"Get more RAM" may be one of your best options. :) Prices are reasonably low right now, and it's one of the best upgrades you can gift your computer anyway. 10k x 10k is borderline but still doable ...
Accepted

### Taking derivative using FFT

I believe this stems from the fact that your function $f(x) = x^2$ does not have continuous derivatives once it is extended periodically like $$\tilde{f}(x) = f(x \ \mathrm{mod} \ 12 -6),$$ which ...
• 2,936
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### What is the advantage of using a particular RK Scheme?

There are lots of different properties which can be found in different time stepping schemes of the same order of accuracy: Different stability properties. While it may not appear that way with the ...
• 2,814
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### Gauss-Lobatto quadrature and nodal points for FEM

Ok, here comes the answer promised in the comment section. Let's start the other way round, going from a general grid to Gaussian grids and further constructions such as spectral elements. In grid ...
• 3,197

### Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods

Indeed the problem was that you were trying to calculate your nonlinear term incorrectly and you forgot that: $$\mathcal{F}[(f(x))^{2}] \neq (\mathcal{F}[f(x)])^{2}$$ I changed your code to this: <...
Accepted

### Need an example Legendre-Gauss-Radau pseudospectral differentiation matrix or Matlab code

This is probably too late to help you, but here is a compact code that will do points, weights and first derivatives for Gauss, Lobatto or either Radau. ...
• 188
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### Spectral methods, Spectral Volume methods, Spectral Difference methods

"Spectral methods" usually means methods which make use of global basis functions. Fourier spectral methods use sine and cosine, and are used when you have periodic boundary conditions. Chebyshev ...
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### Role of weight function in Galerkin methods

Chebyshev are orthogonal wrt to a weight function which is singular at the end-points. When you approximate a function f(x) with Chebyshev, the convergence of the approximations is not affected by the ...
• 3,038
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### Pros of Fourier-Galerkin spectral methods

Pros: With trigonometric basis functions your problem size is $N \text{log}(N)$ instead of $N^2$. Stabilization techniques are easy to implement and cheap: Filtering in the modal space. Zero padding ...
• 1,478
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### Solving ODE with Spectral Method using Chebyshev Polynomials

Your code does not solve the BVP you posted. Here is the revised version that works well. ...
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### What does the Chebyshev differentiation matrix look like for third and fourth derivative?

Yes, you can take the third and fourth power of the Chebyshev differentiation matrix for approximating the third and fourh derivative. Why is this so? Because differentiation is an associative ...
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### orthogonal basis functions on arbitrary domains and boundary conditions

From a mathematical perspective, the eigenfunctions of the Laplace operator span the space $L^2(\Omega)$ regardless of whether you choose Dirichlet or Neumann boundary conditions. As a consequence, ...
• 56.2k
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### Chebyshev/Lagrange polynomials in spectral methods

Your understanding is perfectly fine, except for the last statement that Lagrange polynomials turn out to be a more suitable choice. In fact, both methods, the modal and the nodal expansion, have ...
• 3,197
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### How can we calculate mixed derivatives numerically using the Chebyshev derivative matrix?

Note: Your nomenclature is only valid on Cartesian elements. If you want to calculate derivatives on arbitrary shapes you also have to consider spatial metric terms. Answer: To keep it simple, we ...
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