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You would need to linearize the problem. I prefer to do it before discretization but it's possible to do also after discretization. (I'm a bit skeptical of linearization after discretization because I have never looked into the details. In general, discretization and linearization steps do not commute.) In the following I assume that the equation is actually ...

3

A search for the specific coefficients listed led me to the method ROS3PRL from J. Sieber, Konvergenzanalyse und Numerische Tests für die Prothero–Robinson–Gleichung (Master thesis), TU Darmstadt, 2014. I can't seem to find this thesis online, but the method is mentioned in the following which may be of interest. Rang, Joachim. "Improved traditional ...

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Well, boy, do I have the resource for you :-) Take a look at lectures 26 and 27 at https://www.math.colostate.edu/~bangerth/videos.html

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I'm not going to debug your code, but since $$(1-x)u_{xx}$$ can be discretized, using second order, centered, finite differences with: $$(1-x_i) \frac{u_{i+1}-2u_i+u_{i-1}}{h^2}$$ where $x_i$ is one of the grid points, $h$ your discretization parameter, this means that the $i$-th row of the second derivative matrix is multiplied by the factor $1-x_i$ The $i-$...

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First off, I'd note that your initial condition doesn't satisfy the boundary conditions, so you might want to instead use $u_0(x) = e^{-x^2} - e^{-L^2}$. A great sanity check for problems like yours is the conservation property -- the total mass of $u$ should stay the same. \begin{align} \frac{d}{dt}\int_{-L}^Lu\, dx & = \int_{-L}^L\frac{\partial u}{\...

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As it turns out, the $(u_x)^2$ term is ill-defined due to the irregularity of $u$, which is caused by the space-time white noise. This causes discrete approximations to converge to solutions to different equations, in a similar phenomenon as is discussed here. The solution is to use Martin Hairer's regularity structures, which provides a framework for ...

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