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Are there numerical methods of solving the following fully nonlinear time-dependent PDE: $$\nabla^2u\left(\textbf{r}(t), \dot{r}(t), t\right)=f\left(\textbf{r}(t), \dot{r}(t), t\right),$$ for $\textbf{r}(t)$, where $\nabla^2 = \frac{\partial^2}{\partial^2 x}+\frac{\partial^2}{\partial^2 y}+\frac{\partial^2}{\partial^2 z}$ and $\mathbf{r}(t)=x(t)\mathbf{i}+y(t)\mathbf{j}+z(t)\mathbf{k}$?

Could FACR (Fourier Analysis Cyclic Reduction) be used to solve this?

Also, could you recommend a reference that would help me numerically solve this or a similar problem?

thanks

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  • $\begingroup$ Note that with an iterative method, it can be beneficial to solve for the difference $f(t_2) - f(t_1)$ to go forward in time $\endgroup$ – Jannis Teunissen Jul 4 '16 at 20:07
  • $\begingroup$ @ChristianClason I've clarified: I'm interested in when $u(t)$ is a function containing time-derivatives. thanks $\endgroup$ – Geremia Jul 4 '16 at 20:48
  • $\begingroup$ @ChristianClason Thanks for the welcome. I didn't know this SE existed until today. I'm not sure why it's called computational science, though, which makes me think of cs.SE, and not something like "numerical methods".SE. $\endgroup$ – Geremia Jul 4 '16 at 20:50
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    $\begingroup$ @Geremia Both "computational science" and "scientific computing" have become standard terms for the application of computational methods in the physical (and biological, and engineering, and ...) sciences. (There's a slight difference, if you care to make one, but it doesn't really matter here.) It's definitely not computer science, but it's also not (just) numerical mathematics. $\endgroup$ – Christian Clason Jul 4 '16 at 21:38
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    $\begingroup$ In any case, the answer to your question is a resounding "No, FACR cannot be used, since that is a specific (multigrid) method for solving the linear systems arising from the linear Poisson equation". $\endgroup$ – Christian Clason Jul 8 '16 at 15:24
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To keep things simple, we will approximate $\nabla^2u(\textbf{x},\dot{\textbf{x}},t)$ with some operator $L[u(\textbf{x},\dot{\textbf{x}},t)]$ that uses finite difference approximations:

$$ \nabla^2u(\textbf{x},\dot{\textbf{x}},t) \approx L[u(\textbf{x},\dot{\textbf{x}},t)] = \sum_{k=1}^{d} \frac{u(\textbf{x}-h\hat{\textbf{e}}_k,\dot{\textbf{x}},t) - 2u(\textbf{x},\dot{\textbf{x}},t) + u(\textbf{x}+h\hat{\textbf{e}}_k,\dot{\textbf{x}},t)}{h^2} $$

where $d$ is the dimension of $\textbf{x}$, $h \ll 1$, and $\hat{\textbf{e}}_k$ is the unit vector in the $k^{th}$ direction. Based on this approximation, we can then view the solution to this problem as finding a path based on the constraining "dynamics" you have listed. This can then be viewed from the perspective of Calculus of Variations, where we wish to find an optimal path. This allows us to specify a cost function to minimize, such as:

$$J[\textbf{x}] = \int_{t_1}^{t_2}\left( L[u(\textbf{x},\dot{\textbf{x}},t)] - f(\textbf{x},\dot{\textbf{x}},t)\right)^2 dt$$

where $t_1$ and $t_2$ bound the time frame you care about, and your optimal path $\textbf{x}$ will minimize the cost. To make this path more easily approximated, let's approximate $\textbf{x}(t)$ by some basis, expressed as the following:

$$ \textbf{x}_{h}(t) = \sum_{j}^{n} \textbf{a}_{j} \phi_{j}(t)$$ $$ \dot{\textbf{x}}_{h}(t) = \sum_{j}^{n} \textbf{a}_{j} \dot{\phi}_{j}(t)$$

By using this approximation, $\textbf{x}_{h}(t)$, in place of $\textbf{x}(t)$, the problem can be tackled by finding the coefficients $\textbf{a}_j$ such that the cost function is minimized. This optimization looks like it will be ugly, but this is at least one way you might go about solving it.

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  • $\begingroup$ interesting. I've re-edited my question to making $u\left(\vec{r}(t), t\right)$ any function involving the first derivative of $\left|\vec{r}(t)\right|$. I'm interested in a general solution to such a "time-dependent Poisson equation" (if it can even be called a Poisson equation). $\endgroup$ – Geremia Jul 8 '16 at 2:40
  • $\begingroup$ @Geremia so you mean you're looking for a general approach to solving this problem numerically, correct? Because when I hear the term "general solution", I think closed form (which could be ideal but I think is unlikely). $\endgroup$ – spektr Jul 8 '16 at 3:02
  • $\begingroup$ Yes, I want a numerical solution, not a "closed-form" one. $\endgroup$ – Geremia Jul 8 '16 at 3:10
  • $\begingroup$ @Geremia well I believe this is at least one approach you can take to solve this problem in a general sense. $\endgroup$ – spektr Jul 8 '16 at 3:22

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