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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.

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176 views

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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76 views

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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1answer
88 views

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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0answers
67 views

Finite difference scheme for unconfined aquifer equation

For an unconfined aquifer we have this PDE for the water table position( of course after somehow making the original Boussinesq equation linearized ): $$ \frac{\partial^2(h^2)}{\partial x^2} + \frac{\...
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2answers
385 views

Linear solvers: How to deal with a singular system? (Poisson equation with Neumann boundary conditions)

The question is in the context of iterative numerical solution of large PDE systems with Finite Differences or Finite Elements: Stating the Poisson equation with Neumann boundary conditions will lead ...
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1answer
49 views

Approach for coupled equations with included time derivative

I have an equation system, constructed as $$\begin{split} \partial_tA&=f_1(A)\cdot f_2(B)\\ \partial_tB&=f_3(B)\cdot\partial_tf_4(A) \end{split}$$ with $f_1,\,f_2,\,f_4$ and $f_3$ nonlinear ...
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28 views

Comparison of different methods for solving $\partial_tA=\nabla^2_xA$ [duplicate]

For testing I would like to solve the equation $$ \partial_tA=\nabla^2_xA,\,A_0=\sin(x) $$ In order to solve it, I use an implicit solver $$\left(1-\frac{\nabla^2_x}{2}dt\right)A_{n+1} = \left(1+\frac{...
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1answer
350 views

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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0answers
129 views

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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0answers
34 views

Solving two coupled PDEs using two methods, how to calculate the error between them in theory?

Assumed I have the following coupled PDEs with the variables $A(z,t)$ and $B(z,t)$ and the constants $c_1,\,c_2$: $$\begin{split} \partial_tA&=\partial_zA+c_1\left(A+B\right)\\ \partial_tB&=\...
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1answer
133 views

Ritz, Galerkin, Weak Form, FEM: How to catch up the basics? [duplicate]

I have to deal with FEM and the numerical solution of PDEs a lot. While I'm doing ok when just applying or implementing it, I observe a lack of understanding when authors begin to argue with "Ritz", "...
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0answers
48 views

How to determine which parameter I should linearize?

Assumed I have the following equations with the parameters $\vec{A}(z, t)$, $\vec{B}(z, t)$ and the constants $a,\,b,\,c$: $$ \begin{split} \partial_tA&=a\partial_zA\\ \partial_tB&=b\...
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1answer
53 views

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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1answer
133 views

Implicit method for two coupled PDEs

I have two equations (coupled), with the variables $T_1$ and $T_2$ and the constant $T_0$, which are (when written unitless, i.e. without prefactors): $$\partial_t T_1 = 1-T_1^3+T_1+\nabla\left(\frac{...
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0answers
69 views

Numerically solving generalized eigenproblem with Neumann conditions

I am interested in finding the eigenvalues/eigenfunctions of problems such as $$ \partial_{xx} u = \lambda \partial_{yy} u, $$ which can be solved as the generalised eigenvalue problem $$ \mathbf{A}...
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1answer
353 views

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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2answers
150 views

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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1answer
124 views

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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1answer
73 views

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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2answers
397 views

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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1answer
122 views

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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2answers
335 views

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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0answers
182 views

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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1answer
507 views

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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2answers
83 views

Parabolic differential equations with time delay

Let $d_1=1,d_2=2,a_{11}=\frac{5}{13},a_{12}=\frac{22}3,a_{21}=-2,a_{22}=\frac{6}7,\tau=\frac{5}7$, $\psi(t,x)=\cos^42x,\phi(t,x)=\frac{3}{13}x^4\sin^2 3x$, $\Omega=[0,200]$ How to solve: $$\left\{ \...
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1answer
318 views

Solving an equation in space and time using the Crank-Nicolson approach

Assume I have the following equation (light propagating in $z$-direction through the matter): $$id_zu+d^2_ru=0$$ with $u(z, r)$ being a complex wave. The time scale in this equation is $$t\equiv t_\...
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0answers
87 views

Discontinuos Galerkin Method - inhomogeneous advection problem

I'm currently trying to get into this topic. I've learned that the basic scheme for the advection problem ($D_{x}u+a*D_{t}u=0$) can be solved in a scheme like $$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}...
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0answers
216 views

Stability of nonlinear partial differential equation

I want to find an expression for the stability of the nonlinear Poisson equation. I know about von Neumann stability analysis which applies to linear equations as far as I know. Any suggestion how to ...
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1answer
774 views

Solving an iterative, implicit Euler method in MATLAB

I'm trying to solve an iterative problem that includes an implicit (backwards) Euler method to find successive time values for a given function. The numerical problem is shown here: $$ \begin{...
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1answer
158 views

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 &\...
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2answers
99 views

Trying to compute the error from comparing two arrays

Some context: I am working with the Black-Scholes model.. I have an explicit (Black-Scholes) formula which is the exact solution to my problem. I have written code which implements a finite-difference ...
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0answers
113 views

How to prove the relationship $\|I_h u\|_{0,\infty}\leq C$ for $u\in H^2(\Omega)$?

If $I_h$ is the bilinear interpolation operator, i.e. $I_h: H^2(\Omega)\rightarrow V_h$, where $V_h=\{v_h\in L^2(\Omega)|v_h|_K\in \mathrm{span}\{1,x,y,xy\},~~K \in\mathcal{T}_h\}$ i know that $\|...
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1answer
124 views

Well-posedness of Elasticity Boundary Conditions

For geotechnical engineering problems, it is common to fix a single component of displacement along a boundary as a Dirichlet boundary condition (roller boundary condition). However, I'm having ...
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4answers
261 views

Mass transport in porous media

My goal is to numerically solve the convection-diffusion equation of the form: $$\frac{\partial C}{\partial t} = \nabla (D \nabla C) - \nabla (v C)$$ $C$ is concentration and $D$ is the diffusivity ...
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0answers
125 views

Breather solutions of Sine-Gordon Using Finite Differences

I'm attempting to simulate a standing breather of the form $$ u(x,t)=4\tan^{-1}\left(\sqrt{3}\cos\left(\frac{t}{2}\right)sech\left(\frac{\sqrt{3}x}{2}\right)\right)$$ for the Sine-Gordon equation $$u_{...
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0answers
197 views

How to impose Neumann boundary conditions in finite volume problems?

I'm trying to better understand finite volume methods and have started coding up a basic script to solve the diffusion equation $$u_t = u_{xx}$$ which has the finite volume form: $$\frac{\bar{u}^{n+1}...
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0answers
123 views

Some questions about fractional derivative

I have read some papers about time-fractional PDEs solved by finite element method. The time-fractional derivative is the Caputo derivative defined by $$ \frac{\partial^{\alpha}u}{\partial t^{\alpha}}...
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1answer
189 views

How to construct a function in $H^3(\Omega)$, but not in $H^4(\Omega)$?

As the title mentioned, some suggestions and references are needed to help me to construct a function in $H^3(\Omega)$, but not in $H^4(\Omega)$?The domain $\Omega=[0,1]\times[0,1]$ and has zero trace....
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1answer
149 views

Numerical computation of the velocity in the steady Navier-Stokes equation

I've asked this question on Math.SE too. Let $d\in\left\{1,\ldots,4\right\}$ $\Lambda\subseteq\mathbb R^d$ be bounded, nonempty and open and $\partial\Lambda$ be Lipschitz $V:=\left\{u\in H_0^1(\...
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1answer
496 views

Time discretization of the variational formulation of the Navier-Stokes equation

I've asked this question on mathoverflow too. Let $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):...
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2answers
89 views

A program to simulate cellular automaton model

I have worked on mathematical modeling based on differential equations, and now I want to simulate a cellular automaton based on a ...
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1answer
89 views

If I discretize a PDE in space with WENO and in time with an implicit method, do I need to solve a nonlinear algebraic system at each time step?

I am attempting to solve a nonlinear advection diffusion equation $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x} + u^2)$$ with Robin boundary conditions ...
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1answer
152 views

How to correctly define the flux in a finite volume method for Poisson's equation with a piecewise constant material

How do we correctly define the flux in a finite volume method applied to Poisson's equation where we have a piecewise constant material? Specifically, say we have the equation \begin{align*} -\nabla\...
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1answer
129 views

How do I simulate an open end?

When simulating a partial differential equation describing a physical phenomenon like vibrations on a string, fluid flow in a chamber or quantum wave functions, the most straight-forward way is to ...
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1answer
146 views

Converge rate analysis: issue with time convergence

I have written a code which solves the incompressible formulation of the Navier-Stokes equations. It uses high-order methods both for time and spatial derivatives. I have been conducting convergence ...
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3answers
152 views

When is it safe to ignore the diffusion term in an advection-diffusion equation?

Given the one dimensional equation: $\epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0 $ with $0\le\epsilon \ll1$ with boundary conditions $u(0) = 0$ and $u(1) = 2$, we ...
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3answers
184 views

Numerically solving $\nabla u(x,y) = f(x,y,u)$ on a rectangular domain having initial value of $u$ at some point

Problem description: I want to numerically solve system of two time-independent partial differential equations (pde) of the following simple form $$\frac{\partial u(x,y)}{\partial x} = f_1(x,y,u),$$ ...
9
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1answer
273 views

$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)

I know that the piecewise linear finite element approximation $u_h$ of $$ \Delta u(x)=f(x)\quad\text{in }U\\ u(x)=0\quad\text{on }\partial U $$ satisfies $$ \|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)} $...
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2answers
163 views

Finite element method applied to 1D structural problem - what is wrong with body force?

I have quadratic finite element - shape function is quadratic. Element spans from 0 to 5. Body force is given by (in physical coordinates) $$f_b = \int_0^5 N(x)^T b(x) dx \approx \sum_{i=1}^3 N(\...
3
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2answers
223 views

Why does the displacement have to be small to use the infintesimal strain elasticity equations?

In all the sources I have that discuss the elasticity equations, they start by saying that the strain tensor, $\epsilon$, is related to the displacement gradient, $\nabla u$, where $u$ is the (vector ...