Questions tagged [boundary-conditions]

For questions regarding the choice and/or appropriateness of conditions necessary to model a particular phenomenon with a partial differential equations.

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349 views

Proving solution existence and uniqueness of the Helmholtz equation with Robin boundary conditions with complex coefficients

I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation $$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
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2answers
370 views

Modelling question: example of a physical phenomenon with this jump condition at an interface?

in our finite element class we were talking about interface problems our teacher came up with the following, where $K_i$ are two given functions and $u_i$ is the restriction of the solution $u$ to $\...
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1answer
115 views

deal.ii - ParaView "warp by scalar" of my output is not continuous

During our finite element course, we've solved the linear elasticity problem in 2D on a square (GridGenerator::hyper_cube) with $Q_1$ bilinear finite elements in ...
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1answer
71 views

1D wave equation using Finite difference method MATLAB

I have the wave equation $$u_{tt} = 4 u_{xx}$$ with the boundary conditions $$u(0,t) = u(L,t) = 0\,,\quad x \leq 0 \leq 2\pi \,,\quad t\geq 0$$ and initial conditions $$\begin{align} &u(x,0)=\...
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0answers
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Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?

I am trying to simulate the following equation using FDTD $ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $ subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,...
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0answers
22 views

Advection diffusion equation using Crank-Nicolson with total flux and Diriclet BCs

I am trying to model the 1D advection-diffusion equation: $${\partial c \over \partial t} = D_c{\partial^2 c \over \partial x^2} -u{\partial c \over \partial x}.$$ With Robin boundary conditions that ...
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1answer
109 views

Lagrange multiplier for boundary conditions in pure Neumann problem

I'm trying to solve $-u''=\cos(2 \pi x)$ with boundary conditions $u'(0)=u'(1)=0$ and the constraint $\int_{0}^1 u = 0$ I have to use linear finite elements, so let's assume that I have $M$ degrees of ...
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0answers
43 views

How to apply Neumann boundary conditions in Newton's method [closed]

Suppose that I have a very long and tedious set of differential equations. After discretization, I can get a mapping $f:\mathbb{R}^N \to \mathbb{R}^N$ such that solutions $\phi =(\phi_0,...,\phi_{N-1})...
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1answer
58 views

Manufactured solution to 2d convection-diffusion with homogeneous Robin boundary conditions

I am looking for a manufactured (or analytical if it exists) solution to the 2d boundary-value problem $$\frac{\partial u}{\partial t} = \mathbf{a} \cdot \nabla u + D \nabla^2 u \quad \quad \mbox{in } ...
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1answer
109 views

Wave equation, wave is bouncing off Neuman boundary

Wave equation. Mixed BC. Applied Neuman boundary condition ($\frac{\partial u}{\partial x}\big|_N=0$) to the RHS of the domain You may observe the sharp edge in the middle of the string in the image ...
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1answer
122 views

Implementing Dirichlet BC for the Advection-Diffusion equation using a second-order Upwind Scheme finite difference discretization

i am implementing a Matlab code to solve the following equation numerically : $$ (\frac{\partial c}{\partial t} =-D_{e} \frac{\partial^2 c}{\partial z^2} +U_{z}\frac{\partial c}{\partial z}) $$ with ...
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1answer
141 views

Non-reflective boundary condition

I'm currently solving incompressible Navier-Stokes system of equations with periodic flow and high viscosity. Is there any outlet boundary types that avoids the reflection of flow from the outlet back ...
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1answer
118 views

Computing second derivatives with Neumann boundary condition

I am implementing a finite difference method for a PDE with a Neumann boundary condition. I will simplify my question to a single dimension. Suppose I have a PDE $$\frac{\partial u}{\partial t} = \...
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1answer
510 views

How to solve a second order differential equation (diffusion) with boundary conditions using Python

I am having trouble implementing a model from a publication. Huang, K-L.; Holsen, T.M.; Selman, J.R. Ind. Eng. Chem. Res. 2003, 42, 15, 3620–3625 scihub link: https://sci-hub.se/10.1021/ie030109q I ...
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1answer
67 views

Finite element method with two different Dirichlet boundary conditions

I have the problem like this $$ -\triangle u = f \ \ on\ \Omega \\ u = g_1 \ \ on \ \partial \Omega_1 \\ u = g_2 \ \ on \ \partial \Omega_2 $$ If we choose $$ V_1 = \{ \nu_1 \in H^1 : \nu_1 = 0 \ ...
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1answer
727 views

How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
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1answer
3k views

Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
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1answer
80 views

Displacement field not correct?

Consider the elastic equation $$- \operatorname{div}(C \nabla \mathbf{u}) = \mathbf{f}$$ as presented in step-8. Here $\mathbf{u}$ is the displacement vector, let's consider the 2d case. As you can ...
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1answer
168 views

Weak Formulation of Poisson's Equation for Electrostatics with Surface Charge

I am currently attempting to use FEnICS to solve an electrostatic problem with two materials of different permittivity $\varepsilon_1$ and $\varepsilon_2$ forming an interface: Consider a domain $\...
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0answers
56 views

Constrained optimization for non-linear equations in octaveGNU

I have installed Optim1.6.1 package. I would like to solve a system of equations in non linear finite element analysis using constraints as u=1 at certain nodes. u=0 at certain nodes. Typically I find ...
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0answers
119 views

Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions

Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
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2answers
242 views

How to solve second order coupled non linear differential equations

For a project I am doing, I have to solve the following system of differential equations numerically using my own code: $$ x^2K'' = KH^2 + K(K^2-1) $$ and, $$ x^2H'' = 2K^2H + \alpha H(H^2-x^2) $$ ...
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0answers
24 views

Reynolds boundary conditions

I came across this paper comparing various boundary conditions. I am particularly interested to understand how to obtain the Reynolds boundary conditions (refer to equation 28). $$\left( \frac{1}{c} \...
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0answers
35 views

Insert a boundary condition without removing periodicity assumption

I've to perform a multi-fluid internal flow simulation with a code which intrinsically assumes periodicity on a given direction (say z) on a cartesian grid. Nonetheless, the problem I'm trying to ...
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0answers
50 views

Why is this Advection-convection model with insulating boundary losing mass?

I am trying to model a 1-d advection-convection numerically, using an upwind scheme. I'm using the following equation to calculate the value of internal cells: $$C_x^{t+1} = C_x^{t} + D\frac{\Delta t}{...
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0answers
73 views

Pressure boundary conditions in Stokes Equation in 2D

I am solving the steady-state incompressible Stokes equations in 2D: \begin{equation} \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = 0, \end{equation} \begin{equation} \mu\left[\...
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0answers
91 views

Boundary conditions for an FEM approximation of the Laplace operator

Using FEM, I want to approximate the Laplacian $$u = \nabla \cdot \nabla h \, ,$$ where $h(x,y)$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable. I am using MOOSE ...
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0answers
30 views

Semi-analytical/empirical modelling of wall boundary conditions in advection-diffusion-reaction equation with distributed source

Let's suppose I need to numerically solve a 3D steady-state transport equation of the form $$ \nabla \cdot (\mathbf{u} c) = \nabla \cdot (D \nabla c) - \lambda c + S $$ where $c$ is the transported ...
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0answers
33 views

(FD WENO) Correct symmetry boundary condition for Euler equations

I'm trying to solve 2D Euler equations in axisymmetric formulation with finite-difference WENO scheme. I found some info on high-order boundary conditions for plane formulation (in this thesis, for ...
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0answers
60 views

Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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2answers
5k views

FEniCS: how to specify boundary conditions on a circle inside 2D mesh

I would like to numerically find a mutual capacitance of two stripes of metal on the opposites sides of a cylinder. The problem is obviously a 2D Laplace equation. I would like to find the potential ...
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1answer
141 views

Methodology Suggestion for Wave-propagation Problem using Finite Elements

I want to simulate the propagation of a sinusoidal plane wave in a rectangular domain using Finite Elements Method. First, the wave should propagate through a fluid medium, then it will encounter a ...
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0answers
103 views

How are finite volume method boundary conditions implemented without using ghost-cells?

I'm currently trying to implement my own FVM code in cpp, but when I try to calculate the laplacian of a test function, given by \begin{align}\phi_0=\sin(2\pi x)\sin(2\pi y)\end{align}, I get ...
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0answers
169 views

Poisson equation, stiffness matrix positive definiteness, Dirichlet boundary conditions

I have a question regarding the positive definiteness of the stiffness matrix. Specifically, I believe that it should be positive definite only when at least one Dirichlet point is given, so I would ...
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0answers
43 views

How to solve this boundary value problem which has more unknown than equation on MATLAB

I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
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0answers
48 views

Cauchy problem ill-posed?

Find the solution to the Cauchy problem consisting of the wave equation : $$u_{xx}-u_{yy}=0$$ together with initial conditions: $$ u(x,0)=0,$$ $$u_{y}(x,0)=g(x)$$ for some known initial datum $g$. Is ...
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2answers
172 views

Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?

I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions. So far, I have found the problem solved analytically in one ...
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0answers
203 views

"This DAE appears to be of index greater than 1" daeic12 (line76) error code

Hi I am trying to solve a set of pde converted into ODE and DAE using central finite difference method. I have used the MATLAB 'solve' command to determine the coefficients of fictitious nodes for ...
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0answers
59 views

Integrating a wavelike equation with absorbing boundary conditions

I am trying to numerically solve the following equation: $\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with: $\phi(x, 0) = I(x)$ ...
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1answer
682 views

Imposing periodic boundary condition for linear advection equation - Node problem

I've spent the whole day trying to figure out what is the correct way to impose (and implement) periodic boundary conditions $u(0,t)=u(1,t)$ for all $t>0$ for the simple advection equation $u_t + v ...
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0answers
105 views

A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms

Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit. I am studying the paper by Putnam ...
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1answer
120 views

traction boundary conditions in elasticity

I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My ...
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2answers
391 views

Numerical solution of zero-potential time-dependent Schrödinger equation in 1D

I want to solve numerically the one-dimensional time-dependent Schrödinger equation $$i \psi_t(x,t)=-\frac{\hbar}{2m} \psi''(x,t)$$ My issue is that I don't have the physical background to understand ...
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2answers
90 views

inclined/general Dirichlet boundary conditions

For simpilcity, consider a single quad linear elasticity finite element in 2D. The Dirichlet boundary conditions on node 1 and node 2 are easy to implement and can be handled in the standard way. ...
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2answers
375 views

Finite difference method having a discontinuity

I am trying to understand the FDM which is a widely used method solving differential equations by using approximation below. $$\dfrac{\partial u}{\partial x}=\dfrac{u(i+1)-u(i-1)}{2\Delta x}$$ How can ...
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1answer
76 views

Are spatial boundary conditions required for PDEs discretized with Method of Lines?

As far as I understand, you need to define boundary conditions in time and space to select a unique solution to a PDE and make it solvable. However, in ODEs I only need to specific the initial value ...
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2answers
4k views

FEniCS: separate boundary conditions in normal and tangential direction of mesh boundary

Given a vector-valued PDE, I'd like to enforce the boundary conditions $$ \vec{n}\cdot u = g\\ \vec{n}\cdot \nabla (\vec{t}\cdot u) = 0 $$ on the solution $\vec{u}$. If the boundary happens to align ...
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1answer
64 views

Where could error terms that blow up in SWE come from?

I have been working on a solver for shallow water equations with reflective boundary conditions. I have found that it diverges very fast. As a workaround I noticed yesterday that if I smooth the ...
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0answers
107 views

How to use FEniCS to calculate the electric field of an isolated charged sphere

Initially I thought that this is the kind of question which ought to have already been answered in the form of an example online, but so far I haven't found one. I will admit that I am very new to ...
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1answer
135 views

How to apply Dirichlet boundary conditions to time-dependent PDEs?

Assume the time-dependent linear elasticity equation. Using a finite element discretization we obtain $$M\ddot{u}=Ku+F_\text{ext}$$ where $M$ is the mass matrix,$K$ is the stiffness matrix, and $F_\...

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