CSE, SciComp, HPC et al.

1. Tooling
2. Tool

Computational science and engineering, and scientific-computing materials.

How to verify numerical convergence rates

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- From $|u_h - u| \leq Ch^p$, write  $u - u_h = Ch^p + \mathcal{O}(h^{p+1})$. Transform it to a log-log plot to have the straight line $\log|u_h - u| = p\log(h) +\log|C| + \mathcal{O}(h)$.
- The convergence rate $p$ is the slope. If exact solution is known, just like that; otherwise, choose $u_h$ as a reference solution by making $h$ as small as possible so that we have a fine mesh (high refinement level). Then, follow the steps above.
- Log-log plots usually relate _refinement factor_ x _L2-norm error_.

• For a broader explanation, see here.
• To plot beautiful order triangles, see this discussion.

How to compute L2-norm relative error

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- Consider the rel. error the vector ${\bf e} = \frac{ {\bf u} - {\bf u}_h}{\bf u}$.
- Assume ${\bf u}$ a vector whose $j$-th component is the L2-norm of the mesh quantity over the $n$-dimensional space, i.e. $u_j = \big(\sum_{i=1}^n u_i^2\big)^{1/2}$. Repeat for ${\bf u}_h$, the approximate solution.
- Then, for each mesh node $j$, the $j$-th component of ${\bf e}$ is $e_j = \frac{u_j - u_{h,j}}{u_j}$, for $j = 1,2,\ldots,N$, where $N$ is the number of mesh nodes.
- Finally, we can compute the L2-norm integral error in terms of nodal values by doing $\big( \int_{\Omega} {\bf e}^2 \, d\Omega \big)^{1/2} \approx \big( \sum_{j=1}^N e_j^2 \big)^{1/2}$.

MATLAB help functions of interest

• For full support of functions and operators for SciComp, replace X below by one of the topics listed in the table.

>> help X

Topic	Area
elmat	elementary matrices & manipulation
specmat	specialized matrices
elfun	elementary math functions
specfun	specialized math functions
matfun	matrix functions (linear algebra)
general	general / environment commands
funfun	function functions & ODE solvers
polyfun	polynomials & interpolation
ops	operators & special characters
lang	programming-language constructs
plotxy	2-D plotting
plotxyz	3-D plotting
strfun	character-string functions

Python utility snippets

• Write data to file directly from print.

with open('t.txt', mode='w') as f:
    for i in ['a', 'b', 'c']:
        for j in range(3):
            print(i, j, sep=':', end='\t', file=f)  # stream to file

• Check the result from the shell:

$ cat t.txt
a:0 a:1 a:2 b:0 b:1 b:2 c:0 c:1 c:2

Paraview workarounds

• To make it work on new Apple M1 chips, switch off some vars and call it from the Terminal.
• It worked fine for me on v. 5.10.1.

export VTK_DISABLE_VISRTX=1
export VTK_DISABLE_OSPRAY=1
/Applications/ParaView-5.10.1.app/Contents/MacOS/paraview

Reference: see here.