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However, using a simple separation of variables an instructor demonstrated non-existence of the solution for such problem. You are searching for something nonsense as Wolfgang Bangerth correctly said in his comments, I would show you why. You have this homogeneous equation: $$u^{*}_{tt} = u^{*}_{xx}$$ $$u^{*}(x,t) = \sum_{n=0}^{\infty} a_{n} e^{i(x-t)}$$ ...

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Assuming 1-D and equidistant gridpoints with spacing $h$ and some form of homogenous boundary conditions, we can use $\|\nabla v\|^2\approx -h\sum_{i=1}^nv(x_i)D_2v(x_i)$, where $D_2$ is a finite difference discretization of the Laplacian operator, which is usually some variant of a tridiagonal matrix with values $(1,-2,1)/h^2$ along the sub/main/super ...

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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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Crank-Nicolson is a very good classical approach for parabolic PDE like the heat transfer PDE to which it was originally applied. It is relatively easy to understand and implement so it is often presented in basic courses on numerical methods for PDE. pdepe is also very well-suited to this class of PDE (the second "p" in pdepe stands for parabolic). It has ...

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For any time you want $$\frac{u(t,L+h) - u(t,L-h)}{2h} = K , \quad \star$$Of course $x=L+h$ is not a point in your computational domain, but from the equation above you can get an equation for $u(x, L+ h)$ and substitute in. More precisely, if you discretize in space (with finite difference) from $x_0, \ldots, x_n$, then $x_{n+1} = L+h$ and $x_{-1} = L-h$. ...

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Since this is a question with pretty subjective answers, I'll add a couple to Prof. Bangerth's very good list. the theorem of adjoint/dual operators and spaces is pretty crucial to Computational Science. We know that dual-consistent discretizations of the PDEs can obtain superconvergence which is a nice property. But I think the more commonly used outcomes ...

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You'll get everyone to give different answers to this question, and maybe that's alright. Here are some of my favorite ones: Taylor's theorem that a function (of sufficient smoothness) equals its Taylor expansion plus a remainder term. One can consider the Bramble-Hilbert lemma as a variation of Taylor's theorem, but it has different applications and is ...

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Standard examples of PDE to solve with the typically taught basic discretization methods (Crank-Nicolson et al.) are Transport equations, and other first order equations like Burger's, have often explicit solutions and conservation laws that the numerical methods more-or-less satisfy The heat equation with different boundary conditions and source terms is ...

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If you're interested in modelling any type of PDE within MATLAB, the Partial Differential Equation Toolbox should be able to handle anything you're interested in. The complete documentation for the toolbox can be found here. . A suggested workflow for some simple examples can be found here. The solvable equations via the toolbox are described in detail here....

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