# Questions tagged [pde]

Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.

630 questions
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### Crank-Nicolson scheme in space for advection equation

Consider the equation $$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$, for $t,x\in\mathbb{R}$. I'd like to solve this equation forward in space and backward in time, ...
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### How would you specify mixed boundary conditions for a 2D PDE in the matrix used for finite differences

I have the following PDE in 2D: $U_{x} + U_{xx} + U_y + U_{yy} + U_{xy} = f$ where $f$ is a constant. And I'm trying to create a matrix $A$ to solve the PDE through finite differences: $AU = f$. I ...
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### Need clarification on a piece of book excerpt about spectral element method!

I am reading "Using MPI (3rd edition)" from William Gropp, where in chapter 4 application section 4.13, it introduces an MPI application Nek5000/NekCEM which is based on spectral element method (SEM) ...
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### Crank-Nicolson algorithm for coupled PDEs

Assumed I have the following two coupled equations $$\begin{split} \partial_tA&=a_0AB\\ \partial_tB&=b_0AB \end{split}$$ but I am not sure how to calculate them. One approach is a crank-...
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### Why can I not solve the negative advection equation (backwards in time)?

Suppose we have the negative, inhomogeneous advection equation: $$\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)v(t,x)=u(t,x)\qquad(t\in\mathbb{R}_{+},x\in\mathbb{R})$$...
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### Examples of finding eigenfunctions of coupled DEs

I am looking for examples of numerically finding eigenvalues/eigenfunctions of coupled DEs. If anyone is able to point me towards any examples, preferably with code included, it would be much ...
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### Crank-Nicolson method for inhomogeneous advection equation

Suppose we have the inhomogeneous advection equation $$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$ for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
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### Improving calculation algorithm for coupled PDEs

I have the following two PDEs: $$\partial_zU=\nabla_r^2U+\varrho U$$ $$\partial_t\varrho=a\vert U\vert^4$$ with $a$ a constant and $$dt=dz\cdot\frac{n}{c}$$ with $n$ the refractive index of a ...
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### Stability of PDE Discretizations with Multistep Time Discretizations

Let's pretend we have a spatially discretized PDE of the following form: \begin{align} \frac{\partial^2 \boldsymbol{u}^k}{\partial t^2} = D\boldsymbol{u}^k \end{align} where $D$ can be any form for ...
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### Questions about implementing an electromagnetism/photonics solver package

I am hoping to start (very slowly) on implementing some form of a computational photonics/electromagnetism package. I know things like Meep, S4, FDTD++, EMPy, and a host of other proprietary/free/...
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### Mixed formulation of the Poisson equation (FEM)

I'm solving the Dirichlet problem for the Poisson equation in a 2d domain $D$: $$\begin{cases} \Delta u = 0 \quad \text{in D}, \\ u|_{\partial D} = u_0. \end{cases}$$ I'm interested in ...
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### What's the definition of $L^{\infty}$-norm for nonconforming finite element?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={\text{ess} \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in ...
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### Correctly setting boundary condition for periodic linear elasticity problem

From an old, wise engineering book Peterson's Stress Concentration Factors (http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470048247.html page 324) I've got the following problem: There is 2D ...
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### Implement Robin boundary condition (finite volume)

I have a PDE equation with Robin Boundary condition in an annulus system and I should solve it by finite volume method: \begin{align} \frac{\partial T_f}{\partial t} - k \left(\frac{\...
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### Von Neumann stability analysis with a constant term

I have a question concerning the von Neumann stability analysis of finite difference approximations of PDEs. There seem to be a wealth of online source explaining the application of this stability ...
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### An example of mixed elliptic problem using lowest-order Raviart Thomas element

I try to solve the following mixed second order elliptic PDE in the domain $D=[0, 1]^2$ \begin{eqnarray*} v+\nabla p=&0 \quad &\text{in} \quad D,\\ \text{div}(v)=&1/2 \quad &\...