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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31 views

Linear advection equation: 1D wave with reflection at boundaries (solid walls)

I recently asked a question about this topic, but too much was unclear to me at the time which yielded poor response. (You can find the previous question here). Anyway, I am now trying to solve the ...
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69 views

Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?

I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
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1answer
66 views

FEM diffusion: inaccurate results small time steps

I wrote some FEM code and found some strange results when using a very small time step. So, I decided to analyze the discrete equations. Consider the following linear diffusion problem in 1 dimension:...
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16 views

Error at boundary due to fixed b.c in 3rd order pde

I have a 3rd order in space and first-order in time PDE to solve, as an initial value problem, using the finite element method under fixed boundary condition at endpoints. My endpoints in space are <...
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48 views

Residual of Poisson equation with periodic boundaries

I am trying to write a multigrid solver for Poisson's equation, $-\Delta u=f$, on the unit square, $\Omega=(0,1)^2$ with periodic boundaries. My primary source has been Multigrid by Trottenberg, ...
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44 views

Incorporating radiation boundary condition at the edge in finite difference

I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge. $-k\frac{\partial ...
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1answer
87 views

Simulating 1D diffusion

I'm trying to understand the influence of Neumann boundary condition while simulating 1D diffusion equation $$ \frac{\partial C}{\partial t} = \nabla \cdot (D \nabla C). $$ The initial value is set ...
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32 views

Pressure boundary conditions in Stokes Equation in 2D (Finite Volumes)

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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1answer
49 views

Dealing with boundary conditions using Fourier spectral methods

I am currently working on a project where I need to use Fourier spectral methods to solve the KS equation. I found this code which is using the Fourier spectral methods to solve the classic 1D heat ...
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36 views

How to account for a corner node with zero-flux condition at an extrapolated distance

I am trying to implement a numerical solver and am having troubles dealing with boundary conditions, especially in the corners. I have a 2D mesh, and on the left I have a Dirichlet condition, on the ...
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1answer
89 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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36 views

How to avoid density getting “deleted” in two way rigid body coupling with LBM CFD?

I've been reading this paper recently, which talks about using Lattice Boltzmann methods and two way coupling. Specifically, it outlines fluid solid coupling, and solid fluid coupling, and how simply ...
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45 views

What precautions should be taken when using 2D Perfectly Matched Layers?

I'm solving linearized Navier-Stokes equations with Perfectly Matched Layers in two spatial directions $x$ and $y$, but in the time-harmonic frequency $\omega$-domain, which is meant to be less ...
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45 views

DirichletBC definition on boundary subdomain for component of vector valued function in FEniCS

I am trying to impose a no-outflow condition for a velocity-field over a boundary sub-boundary domain in FEniCS. What I have find challenging is imposing the condition on a component of a vector-...
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2answers
133 views

Is the diffusion equation with Neumann and Dirichlet BCs well-posed?

I am considering the following diffusion equation: $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$ over a grid ...
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70 views

CFD and finite volume method: Dirichlet boundary conditions for the Euler equations

Please point me to an answer if one already exists, but after some searching, I still can't find the answer to what seems like a very simple question. There are plenty of references out there for ...
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38 views

Solving a system of PDEs with no-flux boundary conditions (finite difference discretization)

I am interested in solving a system of linear PDEs with the finite difference method and I'm having trouble to solve the no-flux boundary condition correctly. \begin{align} \frac{\partial n}{\partial ...
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25 views

Interpreting results of using no-flux boundary condition

I am solving for solute transport in 1 D. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$ No-flux boundary condition is imposed at both the ...
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1answer
92 views

Implementing Robin Boundary condition (finite difference)

I'm interested in applying Robin boundary condition to a convection-diffusion problem in 1D. In the following system, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\...
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62 views

Solving diffusion equation using finite difference method

I am solving an 1-dimensional diffusion equation with Neumann boundary condition at outlet and constant concentration, C, at the inlet. In the end, I want to observe how the concentration diffuses ...
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1answer
142 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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1answer
80 views

1-D boundary value problem: How implement mixed boundary conditions using a FD method?

I have been given a convection-diffusion ODE modeling the steady state temperature of a pipe (through which flows a fluid) as $$-\frac{d}{dz}\left(\kappa \frac{dT}{dz} \right)+v\rho C\frac{dT}{dz}=Q(...
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1answer
136 views

Implementing structured grid boundary conditions using NumPy arrays?

I am making a toy code in Python to solve the advection equation $$u_t + cu_x = 0$$ with, for example, periodic boundary conditions. Background information The numerical grid is specified like this: ...
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66 views

How to implement the following Finite Element method for Burgers' equation?

I am trying to replicate this result. It involves using the Galerkin finite element approach onto the viscous Burgers' equation. However, my implementation (in R) seems to be giving me wrong results....
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1answer
90 views

Well-posedness of Navier-Stokes equation

Before running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition?? Can someone ...
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105 views

Implementation of boundary conditions for 1D Euler equations

I'm trying to solve 1D Euler equations with gravity in spherical coordinates using a finite-difference TVD MacCormack method on a non-uniform grid of $N$ components, following the method provided in ...
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0answers
75 views

FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\...
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52 views

A non linear ode with boundary conditions at infinity

I want to solve the non-linear ODE $$\frac{d^2}{dx^2}y=a(y+y^3)$$ With the boundary conditions that $$\lim_{x\to \pm \infty} y(x) =0$$ I am not aware of any analytical method for solving this kind ...
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1answer
150 views

Relation between Stress/Strain and normal derivative of displacement

I calculated the stress $\sigma$ and strain $\varepsilon$ for a solid plate with dirichlet boundary conditions $u = g$, where $u$ is the displacement. With these I want to calculate $\nabla_n u = t$ ...
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42 views

Neumann boundary conditions on arbitrary surface for finite difference diffusion

I am facing the following problem, formulated in practical terms: I have a region $\Omega$ in two or three dimensions, represented as a binary mask, and an initial density $u_0$ within that region ...
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1answer
132 views

Dirichlet boundary conditions in the 1D Heat Equation

Please consider the assignment I have uploaded on the picture. I am confused about the functions $g_L(t)$, $g_R(t)$ and $\eta(x)$. What are they and how do I find them... My question: Is it possbile ...
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34 views

Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme

I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$). $\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$ Where $...
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0answers
41 views

Boundary conditions for solving the time-independent SE for the hydrogen atom

I am trying to solve the schrodinger equation for the hydrogen atom numerically, using finite elements, with matlab's solvepdeeig(). I have a hard time getting the solution to be right, and it seems ...
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1answer
83 views

Imposition of Dirichlet BC for Fourier pseudospectral in this paper

I was trying to implement the algorithm from the paper "Adapting a Fourier pseudospectral method to Dirichlet boundary conditions for Rayleigh–Benard convection". I am having a hard time to ...
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2answers
194 views

Non-linear Boundary Value Problem. How to compute the Jacobian?

Consider a Boundary Value Problem: $$ \delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b $$ $\delta,a,b$ are known parameters. I want to implement Newton'...
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0answers
47 views

Existence and uniquness of solution of FVM for Poisson equation

I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin. $$\Delta u =k,\text{ $k$ constant}\\ \...
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1answer
55 views

Linear elasticity modeling load using traction vs. mixed BC

In classical linear elasticity, when modeling a force/load boundary condition, it appears that we could either: Use a pure Neumann boundary condition, where the 3 traction components are specified. ...
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1answer
49 views

Harmonic oscillators with periodic boundary conditions

I am trying to simulate multiple harmonic oscillators in periodic boundary conditions (subsequently visualizing the process in VMD). I have successfully simulated multiple HOs by using the Leapfrog ...
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1answer
68 views

Structural boundary conditions - rotational/translational DoFs and displacement/tractions BCs

I am a little bit confused over the concept of translational and rotational degrees of freedom (DoFs) in structures, and their relation to displacement/traction BCs. Do displacement boundary ...
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1answer
101 views

Combining multiple coupled 1st order equations in python

I'm having serious troubles with solving translating 3 coupled differential equations into python. The 3 DE's stem from a 4th order DE used to calculate the bending moment of an underwater pipeline ...
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0answers
89 views

PDE discretization on triangular domain

Given the 2D Poisson equation $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$ defined on the domain $\Omega := \{(x,y) \in \...
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1answer
326 views

Discretization Neumann boundary condition

I'm currently working with the following Poisson equation with mixed boundary conditions, including a Neumann boundary condition. $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<...
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1answer
200 views

Heat diffusion - Is this the correct approach to include Newmann boundary conditions?

Thank you for looking at this problem. Is this the correct approach to include neumann boundary conditions? With this solution temperature is not correct, and there´s no diffusion. The model seems ...
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59 views

Unusual boundary conditions on Matlab

I'm trying to solve the following PDE by Matlab, $$ u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T) \tag{1} $$ $$ u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
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0answers
33 views

Hydrogen-like wavefunction as starting guess for atomic solver?

I've been looking into radial solvers for quantum wave equations (Schroedinger and Dirac). In both cases, the suggestion seems to be to go with the "shooting method", with integration schemes of ...
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0answers
73 views

Implementing boundary condition

I'm studying the transport of species A in the blood vessels, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ At x=0, I want to use the ...
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1answer
657 views

Solving advection equation - periodic conditions - using roll python function [closed]

The original post was on stackoverflow : I transfert it here. I have to solve numerically the advection equation with periodic boundaries conditions : u(t,0) = u(t,L) with L the length of system to ...
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0answers
125 views

Are there known accuracy issues between 2D axisymmetric and 3D solutions?

In my full 3D solutions I am solving for the potential throughout a $100\times 200\times 200$ grid. Inside is a ring electrode set to -5V via a Dirichlet boundary condition, and surrounded on all ...
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1answer
390 views

Impose Neumann Boundary Condition in advection-diffusion equation 1D

when solving the advection equation in 1D that is: $$ \frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0 $$ with $ u'(t,0) = 0$ and $u(t,L) = 0$ , $u(0,x) = u_{0} $ one numerical ...
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1answer
1k views

Applying neumann boundary conditions to diffusion equation solution in python [duplicate]

For the diffusion equation $$ \frac{\partial u(x,t)}{\partial t} = D \frac{\partial ^2 u(x,t)}{\partial x^2} + Cu(x,t) $$ with the boundary conditions $u(-\frac{L}{2},t)=u(\frac{L}{2},t)=0$ I've ...

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