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4

There are basically two methods: You can disrectize the angular part via grid points, or you can discretize it via basis expansion. I will focus on spherical symmetry here, the cylindrical case is quite similar. In the basis expansion approach, one applies the ansatz $$v(x,t) = \sum_{klm} a_{klm}(t)\,R_{klm}(r)\, Y_{lm}(\theta,\phi)$$ This is inserted ...

2

It really depends on how the matrix will be used. In implicit schemes, you typically solve a system of the form $(I-\gamma A)x=b$, where $\gamma$ is some small number related to a time step. For anything much larger than 1-D and small 2-D problems, you have to think very hard about how to actually store the matrix and solve the problem. Two popular methods ...

1

Maybe this isn't a helpful response but the reason this happens for the matrix form of Laplacians is because this actually happens for the true infinite-dimensional Laplacians in some settings. In recangular coordinates, $N$-D Laplacians act exactly like their discretized counterparts in this way. For example, let $\Delta_3$ by the Laplacian on $[0,1]^3$ ...

1

This is my attempt at providing some intuition. Everything I state might be obvious, moreover it doesn't have much to do with physics, so this could be a non-answer. I will ignore boundary conditions. Imagine we have a 2D grid with function values $u_{ij}$ for $i=1\ldots m, j=1\ldots n$. Let the $x$ axis point downward, the $y$ axis to the right. N.b. $z$ ...

2

The key feature to a conservative method is simply that the changes due to the fluxes cancel out (i.e., the flux leaving one cell is entering another), so the total mass is constant. Using the form you wrote for a standard conservative method, if we sum $u$ on a grid with $M$ cells, we have: $$\sum_{m=1}^M u^{n+1}_m = \sum_{m=1}^M u^n_m - \frac{\Delta t}{\... 1 There are several libraries for adaptive grids, see e.g., https://math.boisestate.edu/~calhoun/www_personal/research/amr_software/ I have found Petsc to be very useful to write finite difference solvers, even if the schemes are explicit and do not require any matrix solving. The DMDA makes it easy to partition the mesh/solution data, ensure there are ... 4 Let x = x(\xi) be a smooth, invertible map and we make a uniform grid in \xi-space. This induces a grid in x-space$$ x_i = x(\xi_i) $$Method 1: The approximation$$ \frac{F_{i+1} - F_{i-1}}{x_{i+1} - x_{i-1}} = F'(x_i) + O(\Delta x_i) $$is first order accurate as can be checked from Taylor expansion. Method 2: The approximation$$ \xi'(x_i) \frac{...

1

It becomes a root finding problem. Find $u^{n+1}$ such that $$u^{n+1} + \Delta t \theta F(u^{n+1},t^{n+1}) = u^n + \Delta t (\theta - 1) F(u^{n},t^{n}).$$ Now, you can multiply by a test function, form your trilinear form $w(\cdot,\cdot,\cdot)$ to obtain the weak problem; Find $u^{n+1}$ in an appropriate space $V$ such that w(u^{n+1},u^{n+1},v) = (u^n +...

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