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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Are there necessary conditions for SOR algorithm to converge when used for 2D discrete Poisson problem, with PBC, besides solvability condition?

I am trying to solve a 2D poisson problem that is supposed to represent diffusion of chemicals on a grid: $\nabla^2 R_{ij}=f_{ij}$. I am discretizing the problem with the standard central ...
Federica Sibilla's user avatar
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1 answer
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Boundary conditions of a 2D explosion case

I want to solve this problem explosion test case. I was wondering about what are usual the boundary conditions for this type of problems. I want the wave to bounce of the boundary of the domain (let's ...
mle's user avatar
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How to handle non bilinear weak form?

I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment. I modeled the ...
Boiler4562's user avatar
1 vote
1 answer
56 views

How to calculate the force of solid applied by fluid? Using finite difference method, DNS, staggered grid, SIMPLE algorithm, immersive boundary

Problem I am using finite difference method to solve classic problem of flow around cylinder, for validation of my group's immersive boundary method. The common way to validate numerical method is ...
CheapMeow's user avatar
3 votes
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finite difference method not working when going to two dimensions

I have two coupled ordinary differential equations in the steady state: The following code solves, using the Jacobi finite difference method, in 1d using Dirichlet boundary conditions for function $...
BeauGeste's user avatar
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How to correctly implement boundary conditions with Chebyshev differentiation matrix?

In my code I am able to implement zero Dirichlet boundary conditions, however, in my opinion my method does not seem to be as smooth and effective as possible. I am solving a Poisson's equation with ...
Jamie 's user avatar
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1 answer
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How to impose boundary conditions when solving a nonlinear dynamical system given by the FEM solver

I am solving a nonlinear dynamical system given by a nonlinear elastic problem which takes the following form: $$ \boldsymbol{M} \ddot{u} + \boldsymbol{K}_{\textrm{NL}}u = 0 ,$$ here $u \in \mathbb{R}...
Saddam N Y Hijazi's user avatar
3 votes
1 answer
273 views

How to evaluate the points near/at the boundary when using Richardson extrapolation for improved accuracy of a derivative

If we want to improve the accuracy of our numerical estimation of a derivative, we can use Richardson extrapolation. The method is very beneficial when using a centered difference scheme and the ...
FriendlyNeighborhoodEngineer's user avatar
2 votes
1 answer
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What is the correct way to implement Neumann type boundary conditions for solving a PDE using Chebyshev's collocation method?

I am studying spectral methods for solving PDEs numerically. I finished a chapter that explains how to use Chebyshev's collocation method to solve them. Though the explanation in the book is quite ...
Amilox Lex's user avatar
3 votes
1 answer
152 views

Non-standard boundary condition for incompressible Navier Stokes

I am having difficulties applying the boundary condition $$\frac{\partial \vec{V}}{\partial t} + u\frac{\partial \vec{V}}{\partial x} = \frac{1}{\operatorname{Re}}\frac{\partial ^2 \vec{V}}{\partial y^...
2Napasa's user avatar
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Symmetry axis boundary condition

I was wondering about the symmetry axis boundary condition in commercial CFD solvers such as ANSYS Fluent. If the problem is the flow through a round pipe or out of a round nozzle, it is natural to ...
Natalie Leggera's user avatar
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How to set Neumann BC for coupled transport problem in weak form?

Consider $$\begin{aligned} \partial_t v + b\cdot \nabla \phi &=0 \\ \partial_t \phi + b\cdot \nabla v &= 0 \end{aligned}$$ for $v:(x,y)\mapsto \mathbb{R}$, unknown and time-dependent ($...
l'étudiant's user avatar
2 votes
2 answers
167 views

Starting configuration for Molecular Dynamics

I have to build a bilayer using a structure obtained in a previous MD simulation. In the structure, some chains of the molecules cross the simulation box as shown below. I added water layers to the ...
Ema 's user avatar
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2 votes
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Symmetrization of Laplacian Matrix Operator (finite volumes)

The aim is to construct symmetric Laplacian matrix $L$ for 2D square domain. I have the following discretization of second derivatives on non-uniform grid (will skip the steps of derivation, any ...
2Napasa's user avatar
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3 answers
258 views

Dirichlet condition in finite element method

I'm trying to understand the approach described in these questions: How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices How to apply non zero ...
Jake1234's user avatar
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First derivative central differences with reflecting boundary conditions

I have the following problem in 1-D: \begin{equation} \partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega. \end{...
lightxbulb's user avatar
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How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?

I am applying a central Finite Difference scheme in space and an implicit Euler scheme in time on a variant of the 2d Burgers equation, of the form:$$u_t + uu_x + uu_y = \nu(u_{xx} + u_{yy})$$ where $...
Robby Ram's user avatar
3 votes
1 answer
580 views

Solving Poisson's Equation with Periodic Boundary Conditions

So, I've been attempting to design a simple solver for a problem of finding the gravitational potential of a system using Poisson's equation (let's call the potential phi, $\phi$). The goal is that I ...
TheAkashain's user avatar
1 vote
1 answer
213 views

Crank Nicolson Method with closed boundary conditions

I want to simulate 1D diffusion with a constant diffusion coefficient using the Crank-Nicolson method. $$\frac{\partial u (x,t)}{\partial t} = D \frac{\partial^2 u(x,t)}{\partial x^2}.$$ I take an ...
Jbag1212's user avatar
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1 answer
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How can we use the two boundary conditions for the Taylor-Maccoll Equations for Cone Shock Waves?

The Taylor-Maccoll equations below govern the gas dynamics around a supersonic, axially oriented cone. Using the notation from Anderson's Fundamentals of Aerodynamics, which uses a dimensionless ...
Jacob Ivanov's user avatar
2 votes
1 answer
247 views

Why do we use modified pressure in incompressible multiphase solvers with gravity?

The context of my question is two-phase incompressible solvers such as interFoam in OpenFOAM, but I have seen this trick used ...
Robert Manson-Sawko's user avatar
0 votes
1 answer
52 views

Why is the maximum potential energy greater than the maximum kinetic energy?

I was plotting the energy variation in a mass-spring system. If I define the initial conditions to be at maximum displacement from the origin, the potential energy is plotted correctly but kinetic ...
Belal Bahaa's user avatar
1 vote
0 answers
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How to include zero flux boundary conditions?

I am trying to solve the following differential equation in the domain of $\theta \in [0, 2 \pi]$ using finite differences scheme: For $0< \theta \leq \pi$ \begin{align} \rho_i^{n+1}=\rho_i^{n}+D\...
Irbin B.'s user avatar
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Interface condition for 1D Helmholtz equation using finite element method

I want to implement a 1D Helmholtz equation with jump condition. The domain is $x=[0,1]$ and both ends have Dirichlet boundaries($p$=0). The 1D strong formulation is; $$c^2\nabla^2p + w^2p=0 \qquad \...
Ekrem Ekici's user avatar
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boundary condition at rotation axis of a spherically-symmetric system

The quantity I am interested in is not the rotation rate $\Omega$, but I will use this quantity nonetheless to make the problem clearer. I am interested in a spherically-symmetric system and in the ...
BitterDecoction's user avatar
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60 views

Why do BVP solvers' APIs only allow "unknown" parameters in the derivative and residual functions but not "known" parameters?

I recently needed to solve a second order boundary value problem and noticed that both scipy.integrate.solve_bvp and Matlab's ...
user9794's user avatar
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2 votes
0 answers
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Compact Finite Differences for the Heat Equation with Robin Boundary Conditions

I am trying to solve the Heat equation with Robin Boundary condition: $$ u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$ for $ 0\leq x\leq1$ ...
Jules's user avatar
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1 answer
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Nonlinear Robin boundary condition involving square root

If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition at $x=a$ and the Dirichlet ...
mathemania's user avatar
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Want to know how to solve this high order differential equation system with boundary condition?

The numerical results may be like this:
Johny Chou's user avatar
1 vote
1 answer
133 views

Free Time Dependent Schrodinger Equation with Inhomogeneous Dirichlet boundary

There exists a FFT-based method to solve the poisson equation in inhomogeneous Dirichlet boundary condition using the sine-transform. For example, Which fourier series is needed to solve a 2D poisson ...
WhatsupAndThanks's user avatar
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2 answers
221 views

Practical implementation of the discrete compatibility condtion

I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form $$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
Akhaim's user avatar
  • 83
2 votes
0 answers
129 views

Boundary conditions for compressible Euler equations

I want to solve the compressible 1D Euler equations numerically. Theory says that for subsonic inflows, one can prescribe two variables, e.g. pressure and temperature. Density can then be computed ...
DozerD's user avatar
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What EXACTLY are the "outflow" and "pressure outlet" BCs in ANSYS Fluent?

I am just starting my MSc and we have been given an assignment in which we must model two-dimensional, laminar, incompressible Poiseuille flow in a channel. We are supposed to use a VELOCITY INLET ...
K.defaoite's user avatar
1 vote
0 answers
73 views

How to incorporate a continuity of 1st derivatives at an internal interface in a 2nd order Poisson equation in finite elements method?

How should I modify the following matrix formulation of a finite element (FE) approximation of Poisson equation at the (internal) interface denoted by the nodes $i-1$ to $i+1$ having a continuous 1st ...
user28726's user avatar
1 vote
1 answer
49 views

Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions

I'm trying to model some two-dimensional waves, and am unsure how to combine my boundary conditions with spectral methods. The PDE I'd like to explore resembles the equations for damped, driven ...
MRule's user avatar
  • 153
2 votes
0 answers
108 views

Numerical precision on tricky coupled nonlinear boundary value problem on infinite interval

I am trying to solve with high precision the following coupled system $(f,h)$ on $[0,\infty]$: $$-h''-\frac{1}{r}h'+\lambda_c h(f^2-1)+4\lambda_h h^3=0$$ $$-f''-\frac{1}{r}f'+\frac{1}{r^2}f+ f(-\...
phyphy's user avatar
  • 33
1 vote
1 answer
188 views

Boundary value problem solver fails on trivial case

I am trying to solve a boundary value problem on $[0, \infty]$, using scipy's scipy.integrate.solve_bvp and I am seeing that the solutions are not converging even ...
phyphy's user avatar
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0 votes
1 answer
197 views

Can successive over-relaxation (SOR) method deal with ill-posed PDE BVP?

Recently, I've been struggling to understand the limitations and capabilities of the successive over-relaxation (SOR) method for boundary value problems which are ill-posed, such as, for instance, ...
Akhaim's user avatar
  • 83
1 vote
1 answer
286 views

Convergence stall when solving 2D Poisson PDE with pure Neumann boundaries (finite differences)

I recently started coding a small library of 2D PDE solvers (time dependent and time independent), and my first attempt was a 2D Poisson equation of the form: $$\nabla(\epsilon\nabla\varphi)=\nabla\...
Akhaim's user avatar
  • 83
5 votes
1 answer
288 views

Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?

If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$): \begin{alignat}{3} \Delta u(x) &= f(x), &\quad&x\in\Omega \\ \partial_nu(x) &= 0, &\quad&...
lightxbulb's user avatar
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1 vote
1 answer
85 views

How can I define an equipotential surface/volume in FEniCS?

I want to solve electrostatic problem for potential. Charge density and medium permittivity are known, so is the potential of a grounded surface. I know how I can implement that. But I would like to ...
abukaj's user avatar
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1 vote
0 answers
32 views

How does Tannehill impose boundary conditions when coding the Parabolized Navier Stokes on an Implicit Finite Differences Scheme?

I'm trying to implement the scheme he describes on his book "Computational Fluid Mechanics and Heat Transfer" on Chap.9 and I'm having trouble imposing BC. I don’t get how he imposes them. I ...
Tinucci's user avatar
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1 vote
1 answer
106 views

Applying Stress Boundary Conditions in Commercial Finite Element Analysis Codes

I am trying to replicate a finite element analysis given in a research paper titled On the Detection of Stress Singularities in Finite Element Analysis 1 by G.B.Sinclair et. al. The geometry of the ...
Ali Baig's user avatar
1 vote
0 answers
44 views

Adding wall friction to a 3D model - Stokes equation

Maybe that is not the best place for the following question, if such, let me know and sorry. I would like to model the flow of expanding bentonite in a fracture following the given reference. In order ...
Daniel's user avatar
  • 99
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0 answers
684 views

The mathematical meaning of a zero gradient pressure boundary condition in the Navier-Stokes equations

I would like to solve the Navier-Stokes equations for the unsteady problem of the flow around a circular cylinder. I would like to understand how to write mathematically the boundary condition for the ...
Saddam N Y Hijazi's user avatar
0 votes
2 answers
418 views

Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?

I'm solving a local instability problem for a pipe Poiseuille flow. The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial). The basic flow is $\bar{u_r}=0, \bar{u_\...
Jack's user avatar
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1 vote
0 answers
256 views

Open boundary condition for 1d wave equation with variable wave speed using finite differences

I have implemented a finite difference solver for the 1d wave equation with variable wave speed: $$ u_{tt} = c(x)u_{xx}, \hspace{10mm}c(x) = \dfrac{6 -x^2}{2} \hspace{5mm} $$ on $-2 \leq x \leq 2, t &...
Adam Lau's user avatar
0 votes
1 answer
1k views

Who can explain the minimum image convention in molecular dynamic simulations?

How to choose the cutoff radius so that the atoms do not interact with its periodic image? Especially when simulating macromolecules (proteins).
Muhriddin Mahkamov's user avatar
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0 answers
67 views

Passing boundary conditions to solver

Quite broad question, Currently building own Poisson solver subroutine for CFD solver. Works smooth, the goal is to generalise the input and make it flexible. Description of also: Memory allocation. ...
2Napasa's user avatar
  • 362
2 votes
3 answers
141 views

Simple to program method for elliptic PDE with curved boundary?

I know very little about numerical PDE, so I apologize if this equation is ill-formed (and I appreciate any corrections). I am currently learning about Brownian motion. A classic result is that we can ...
alligator's user avatar
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