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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0answers
25 views

Interface Formulation at Finite Volume Boundaries when using the Dual Mesh

When using the dual mesh (vertex-centered) for finite volume methods, you end up with a cell center at the boundaries between materials. It is possible that the equations being solved in each ...
1
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1answer
121 views

Euler's equations 1d for pipe, Inlet boundary conditions

$\def\rmin{{\mathrm{in}}}$ $\def\l{\left}\def\r{\right}$ $\def\tagl#1{\tag{#1}\label{#1}}$ I am using the one-dimensional finite volume method to calculate the air flow in some tube. For subsonic ...
3
votes
0answers
35 views

Discretizing boundary conditions for vortex methods

I am working on a fluid simulation using vortex methods. For this I must compute the vortex sheet on my boundaries given as: $$ \gamma(\mathbf{x}) - \frac{1}{\pi}\int_S ...
1
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0answers
14 views

What type of boundary condition(s) do I need to define for a process with diffusion of tracer concentration?

I want to simulate the diffusion of initial dye concentration in a small estuary with a 2D depth averaged model. Would a radiation boundary condition for concentration of tracers alone be enough or do ...
0
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0answers
22 views

Boundary layer in a stream function

i've to solve numerically this equation: where: psi_y=u psi_x=-v that is the boundary layer equation in a stream function. i want to solve this equation with a direct method. which are the ...
4
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1answer
57 views

How to handle inflow and outflow boundaries for a non-linear convection-diffusion equation (DGFEM)

Following "A conservative DGM for Convection-Diffusion and Navier-Stokes Problems" (Oden and Baumann), if we have a linear convection-diffusion equation of the following form: $$ ...
4
votes
2answers
77 views

Solving system of differential equations with interconnected boundary conditions

I am trying to solve the following system of differential equations numerically over the domain $x=0$ to $x=D$. The main difficulty is that the boundary conditions are interconnected and depend on the ...
3
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0answers
81 views

Transport Equation in a Tube: Source Term on Boundary

I'm modeling mass transport in a flow reactor. The flow reactor is a tube, which allows me to use cylindrical symmetry in solving the Convection-Diffusion-Reaction (CDR) Equation, which governs the ...
3
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1answer
131 views

Problem in Discretizing Convection-Diffusion-Reaction equation

I'm trying to solve the Convection-Diffusion-Reaction (CDR) equation on a rectangular domain, using cylindrical coordinates and Finite Difference Methods (FDM) (this approximates a flow reactor). ...
2
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1answer
154 views

Implementation of gradient zero boundary conditon in advection-diffusion equation

My question is about Finite Element Method. I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like, $\frac{\partial \rho}{\partial t} + ...
2
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1answer
70 views

Boundary conditions for second order PDE

For a second order PDE, for example heat conduction equation $\frac{\partial T}{\partial t} = \frac{\alpha}{C_p} \nabla^2 T$, is it possible to determine the steady-state (or even transient) solution ...
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0answers
41 views

how to find outward normal for robin codition [duplicate]

I wrote the code to fix this problem; now I should validate it using a test function to see the error in my code. I can't figure out how starting from u can derive the function g on the edge, in ...
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1answer
117 views

Pde problem with robin boundary condition

I have my pde 2D problem with robin condition (form: du/dn +ku=g) to solve with matlab. i have the exact function u and I want to find the function g in robin condition. How can i do it? thanks for ...
2
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2answers
73 views

How to impose a constant constraint PDE

What is the best way to impose a "constant constraint" for a PDE? Specifically, I want to solve an eigenvalue problem $Au=\lambda u$ on the rectangle $(0,2\pi)\times(-\pi/2,\pi/2)$ with periodicity ...
3
votes
1answer
115 views

The Lax-Milgram Lemma in FEM with non-homogenous Dirichlet BC

How can show that the prerequisites for the Lax-Milgram Lemma holds if I have different test and trial spaces (which I think is the natural thing to have if at least part of the boundary is ...
4
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1answer
113 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
9
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3answers
296 views

How to deal with curved boundary condition when using finite difference method?

I'm trying to learn about numerically solving PDE by myself. I've been beginning with finite difference method(FDM) for some time because I heard that FDM is the fundament of numerous numerical ...
2
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0answers
50 views

Finite Difference for Hamilton Jacobi Belman

I have hjb equation where $V=V(x,t)$ and $u=u(x,t)$ $V_t + \sup(u) [A(x,u)V_x + B(x,u)V_{xx}]=0$ for $x$ in $[0,1]$ and $t$ in $[0,1]$ I have been able to successfuly resolve it numerically having ...
2
votes
1answer
147 views

How to implement boundary condition in this case?

Let's start off by NUMERICALLY solving a 1-D steady-state heat transport problem using IMPLICIT FDM. $DT_{xx}=0; ~T(x=0)=T_{BL}; ~T(x=l)=T_{BR}$ where $D$ is diffusion; $T$ is temperature; subscript ...
6
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2answers
280 views

Solid mechanics with finite differences: How to handle “corner nodes”?

I have a question concerning coding boundary conditions for solid mechanics (linear elasticity). In the special case I have to use finite differences (3D). I am very new to this topic, so perhaps some ...
3
votes
1answer
87 views

boundary oscillations with Robin boundary conditions

When solving Poisson's equation on the unit square $\Omega$ with homogeneous Dirichlet boundary conditions for $x=0$ and Robin-type conditions at the rest of the boundary, $$ \begin{cases} -\Delta u = ...
6
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3answers
424 views

Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
4
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1answer
242 views

boundary conditions for electrostatic problem

I'm solving an electrostatic problem governed by Laplace equation $$-\nabla \cdot (\rho^{-1} \nabla u) = 0$$ in the following domain: a brick ($\Omega_1$) with a cylindrical inclusion ($\Omega_2$), ...
5
votes
1answer
124 views

What Linear Equation Solver should be used for a problem with many dirichlet conditions?

I am solving a laplace equation on a finite-element mesh (tetrahedral, triagonal) and have many say 99% dirichlet conditions compared to the number of unknowns. Is there an efficient way to solve this ...
0
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0answers
292 views

solving a set of coupled differential equations in matlab

plot h vs t for the below coupled equations: $a_1\frac{d^2p}{dx^2}-b_1\frac{d^2g}{dx^2} = c\frac{dh}{dt}$ $a_2\frac{d^2p}{dx^2}-b_2\frac{d^2g}{dx^2} = c\frac{dh}{dt}+d\cdot g$ where, a1,b1,a2,b2 ...
5
votes
2answers
279 views

Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?

I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" ...
3
votes
1answer
634 views

Finite Difference Method Neumann Boundary Condition with Variable Coefficients

Disclaimer In the process of typing up this question, I determine its solution. Since I went through the trouble of typing up the question in its entirety, I will post its answer as well. It may ...
13
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1answer
1k views

How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the ...
5
votes
1answer
183 views

Are we free to choose the position of ghost cells on a non-uniform finite-volume mesh?

Following Hundsdorfer approach the finite volume discretisation of the advection-diffusion equation (conservative form) on non-uniform cell centered grid can be written as, $$ w_j^{\prime} = ...
3
votes
1answer
148 views

Implementation of a contraction force in Fenics

This might be a bit of an odd question, and sorry about the poor wording im having trouble conceptualizing what I want in my head let alone get it out in a way others can understand. Is there any way ...
0
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0answers
28 views

Computation of potential flow using dirichlet conditions [duplicate]

I know I already posted this question and I thank you for your answers, but unfortunately I didn't find what I was looking for among them. Anyway now I'll rewrite the question more clearly, because ...
5
votes
1answer
174 views

Potential flow around a non-symmetric obstacle using stream functions

I've seen that there is a way to use the finite differences method, on a cartesian orthogonal grid, to perform calculations on potential flow about an obstacle without using the Neumann conditions, ...
6
votes
2answers
466 views

FEniCS: boundary conditions for electrostatic problems with dielectrics

I carefully read all circa 70 pages of FEniCS tutorial and I still do not understand how to solve electrostatic problems when I have materials with different dielectric constant. The self contained ...
2
votes
1answer
645 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 ...
3
votes
1answer
181 views

CFD (Fluent) define a inlet for a tidal basin

I'm still pretty new in the CFD modelling world. Can anyone advise me how to define a inlet for a tidal basin in Fluent? The water level and the velocity at the inlet vary in time due to the tide ...
4
votes
1answer
301 views

Trouble implementing Neumann boundary conditions because the ghost points cannot be eliminated

Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question. I ...
4
votes
1answer
343 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 ...
5
votes
1answer
435 views

Example of a PDE model with nonlinear Dirichlet boundary conditions

Is there any application for PDEs with nonlinear Dirichlet boundary conditions? That is, I am looking for an example of a partial differential equation for a state $u$ posed on a domain $\Omega$ with ...
2
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0answers
283 views

Newton Iteration method convergence

I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
2
votes
1answer
318 views

Role of boundary conditions (e.g. periodic) in Poisson equation

Given 3D Poisson equation $$ \nabla^2 \phi(x, y, z) = f(x, y, z) $$ and the right hand side and the domain, am I free to impose any boundary conditions (BC) on the function $\phi$, or do they have to ...
5
votes
2answers
343 views

No flux boundaries for mixed hyperbolic parabolic PDE

I read this post, "Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation" and although it is the same type of equation it does not fit ...
3
votes
3answers
499 views

Open boundary conditions with the advection-diffusion equation

Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term), $$ \frac{\partial \phi}{\partial t} = ...
13
votes
1answer
883 views

Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation

I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle ...
3
votes
1answer
137 views

Marker and Cell Method (MAC) - STOKES FLOW - boundaries?

please can you help me with my problem with Stokes flow written using Marker and cell method (MAC)? I need only to solve the eq. of continuity + momentum eq. for a given condition (steady state). I ...
4
votes
1answer
170 views

mathematical statement of “open” boundary condition

For your information, the original equation comes from here. Note: You DON'T have to read the paper. I will make the question as self-contained as possible. The central equation to solve is equation ...
7
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2answers
3k views

Writing the Poisson equation finite-difference matrix with Neumann boundary conditions

I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
5
votes
3answers
150 views

which numerical method for ode with mixed BCs

I've got a second order nonlinear ODE (nothing fancy), but the BC are a little odd to me: $y'(0) = 0$ $y \rightarrow y_a$ as $x \rightarrow \infty$ What's a good numerical method for solving ...
5
votes
1answer
772 views

How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
5
votes
1answer
154 views

how to visualize lattice with periodic, helical, etc. boundary conditions?

I am trying to write a special hexagonal lattice generator, with several kinds of boundary conditions, such as helical BC, periodic BC, and I find it hard to verify whether it works correctly. I tried ...
4
votes
1answer
152 views

How do I solve an ODE Two-Point Boundary Value Problem?

I have a feeling my question is a very basic one, but I am not at all well versed in computational sciences. My equations are of the form: $$ y \in \mathbb{R}^3 \\ \dot{y}(t) = f(y(t)) \\ y_1(0) = a ...