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5

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

Based on the image that you provided in your comment, I believe you formulate your problem as a system of PDEs for each branch in your network and make sure at each connecting node mass is conserved. Let's say you have $N$ branches, so you need to solve the system of advection-diffusion equations for each branch ($1 \leqslant i \leqslant N$): $$\frac{\... 2 As I wrote in my comment, you have plotted your errors as a function of time, and asked how that relates to the spatial error of the different finite difference methods you've used, that is not the proper way to measure this. The proper way would be comparing your two methods error to the closed form result for all times and seeing how refining the mesh ... 1 As the mathematica.se thread shows, the solution of$$ \begin{aligned}\frac{\partial}{\partial x}\left( \operatorname{sign}(x) u(x) \right) + \frac{\partial}{\partial x} \left( \sqrt{u(x)} \frac{\partial u}{\partial x}(x) \right) &= 0 & &\text{in } \Omega = (-6,6), \\ u &= 0 & &\text{on } \partial \Omega = \{-6,6\} \end{aligned}$$is ... 1 Since your trial and test spaces are different, you have to use a different version of Lax-Milgram lemma, see e.g., [1], Theorem 5.1.2 You can still use lifting idea since the PDE is linear. Then you can verify the conditions in standard Lax-Milgram lemma. To show coercivity, you need the condition$$ \gamma(x) - \frac{1}{2} b'(x) \ge -\eta, \qquad -\infty &...

1

Take the first expression and start to reduce the $x$ values to $x_i$, \begin{align} \frac{u_{i+1}^n - u_i^n}{x_{i+\frac{1}{2}}} - \frac{u_i^n - u_{i-1}^n}{x_{i-\frac{1}{2}}} &= \frac{(x_i-\frac12Δx)(u_{i+1}^n - u_i^n) - (x_i+\frac12Δx)(u_i^n - u_{i-1}^n)}{x_{i-\frac{1}{2}}x_{i+\frac{1}{2}}} \\ &=\frac{x_i}{x_i^2-\frac14Δx^2}(u_{i+1}^n - 2u_i^n + u_{...

1

I think that the second boundary condition equation is incorrect. The first one should be right for both ends. Following your notation, the flux of mass in the domain should be: $$N = vC - D \frac{\partial C}{\partial x}$$ everywhere including the boundary points. Keep in mind that the sign of the boundary condition value is positive if the direction of ...

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