Contribution

[Christina-wg]

This issue is used for describing and updating the contributions to Pyclaw.

Contribution idea 1:
We intend to either test the performance of different solvers or make a new example on a specific case of the 2D conservational law system
$$u_t + f(u)_x + g(u)_y = 0.$$
Consider the Euler equations for compressible gas dynamics [1]

$$u = \begin{bmatrix} \rho \\ \rho v \\ \rho w \\ E \end{bmatrix}, \quad f(u) = \begin{bmatrix} \rho v \\ \rho v^2 + p \\ \rho vw \\ v(E+p) \end{bmatrix}, \quad g(u) = \begin{bmatrix} \rho w \\ \rho vw \\ \rho w^2 + p \\ w(E+p) \end{bmatrix}, $$

with total energy $E = \frac{p}{\gamma - 1} + \frac{1}{2}\rho (v^2 + w^2),$
where $\rho$ is the density, $p$ is the pressure, $(v,w)$ is the velocity, and $\gamma=1.4.$
Let the mean flow have $\rho=1, p=1, (v,w)=(1,1),$ and we add to it an isentropic vortex with perturbations. The velocity and the temperature $T = p/\rho$ has perturbations, respectively,
$$(dv,dw) = (1,1) + (\frac{\epsilon}{2\pi}e^{0.5(1-r^2)}) (-\bar{y},\bar{x}), \quad dT = -\frac{(\gamma-1)\epsilon^2}{8\gamma \pi ^2}e^{1-r^2}.$$
The entropy $S = p/\rho ^{\gamma}$ has no perturbation.
We use $(\bar{x},\bar{y}) = (x-5,y-5), r^2 = \bar{x}^2 + \bar{y}^2, \epsilon=5,$ and periodic boundary conditions for the domain $[0,10]\times[0,10].$
The initial conditions for the problem would then be

$$\begin{align}(v_0,w_0) &= (1,1) + (dv,dw), \\\ T_0 &= 1 + dT, \\\ S_0 &= 1.\end{align}$$

The goal is to solve the system in Pyclaw with SharpClaw and classic Clawpack packages, and compare their performances.

Contribution idea 2:
We are also considering implementing the 2D Burgers' equation
$$q^t + \frac{1}{2}(q^2)_x + \frac{1}{2}(q^2)_y = 0$$
with piecewise constant initial conditions and period boundary conditions in Pyclaw and making a new example on it.

Reference:
[1] Zhang, R., Zhang, M., & Shu, C. (2011). On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes. Communications in Computational Physics, 9(3), 807-827. doi:10.4208/cicp.291109.080410s

[MaYeatCo]

Text for an issue in PyClaw:

"Hello, PyClaw developers. We are 2 students, in the numerical PDE class led by Jed Brown. We have a project to contribute to the Numerical PDE community, and we are considering adding another notebook/tutorial/example to the gallery or https://github.com/clawpack/apps/. Right now, we focus on using PyClaw to solve the Euler system described in "On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes" by Zhang et al (specifically, example 2.2). Another equation would be Burgers' equation variant, again specifically for PyClaw.

We are slowly getting to know PyClaw better via the website. We tested some examples. Still it's our first time contributing to any package in general, We are open to suggestions/directions (e.g. which equation might be another good candidate)!

P.S. if it's not the right place for this type of communication, my apologies. Please again direct where this should be relevant

Madi Yerlanov and Christina Wang"

[jedbrown]

Cool. I would link the paper by DOI and say specifically that you want to set up example 2.2 (isentropic vortex) as a convergence study. I would focus on comparing SharpClaw (high-order "WENO") with classic Clawpack (2nd order) in terms of accuracy and simulation cost. If that goes well, you might add a problem with a shock. I don't think it's necessary to compare with arbitrary other packages, though you'll be able to run this problem out of the box using ceed-fluids (uses a finite element method), reporting execution time and errors. If you add a shock, ceed-fluids will likely do poorly (there is one shock capturing method that might work).

[MaYeatCo]

Posted ! clawpack/pyclaw#714