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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4
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0answers
1k views

2D wave equation with Mur boundary condition - setting up the RHS and solving (time-steps)

I am trying to solve a 2D wave equation implicitly using FD with central approximations with the following boundary conditions $$\begin{align} &u=2\sin\left(\frac{2\pi}{5}t\right)\quad \text{at }...
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1answer
1k views

Instability of pdepe in Matlab… boundary conditions?

here is a Matlab beginner banging his head on the wall... I am trying to solve a system of partial differential equations in Matlab, with both derivatives in time and space domains. I am using the ...
3
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1answer
347 views

Boundary conditions in conforming Galerkin method for biharmonic equation

I am trying to solve simple scalar biharmonic equation using bubnov-galerkin finite element method. I am using $H^2$ conforming basis functions. I was wondering that if anyone can give me some ...
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1answer
860 views

Solving PDE with state and time dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
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1answer
36 views

In the method of weighted residual, is it necessary for the basis function to satisfy the boundary conditions?

In the method of weighted residual applied to boundary value problems, is it necessary for the basis function to satisfy all of the boundary conditions? Will it work even if it does not satisfy all of ...
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0answers
471 views

Boundary equations for constant right hand side in Poisson equation (FD)

I am setting up to solve a 2D Poisson equation $\nabla^2u = T$ by the finite difference method. I am using the multigrid method. On the coarsest mesh, where the grid size is 2x2, I want to set up ...
3
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2answers
2k views

How to solve ODEs with constraints using BVP4C?

I am using BVP4C to solve a system of ODEs which is as follows. \begin{equation} \left\{ \begin{aligned} \frac{\partial f(x,y)}{\partial x} &- \frac{d}{ds}\big(\...
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1answer
544 views

Shallow Water Equations Boundary Conditions

I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example (http://loki.udc.es/info/asignaturas/calculo_ii/Finite%...
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1answer
492 views

Surface charge boundary conditon for Poisson-Boltzmann solver

I want to set up a surface charge boundary condition for the simulation of semiconductor and electrolyte interface. The 2D-Poisson example and surface charge boundary condition are shown in following ...
4
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2answers
576 views

Manufactured solution for pressure based 3d incompressible Navier-Stokes solver with wall boundaries

I already successfully verified my solver (SIMPLE-type FVM-method) with the following manufactured solution (3d Taylor-Green vortex) on the solution domain $[-1,1]^3$ with Dirichlet boundary ...
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1answer
739 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
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134 views

Arbitrary Choosing of the Solution Domain - Navier Stokes and Manufactured Solutions

I want to verify a finite-volume solver (SIMPLE-Algorithm) for the incompressible Navier-Stokes equations by using a manufactured solution. I use Dirichlet boundary conditions for the velocity at all ...
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1answer
86 views

methods for a peculiar BVP system

Consider the following system defined on the open interval (-1, 1): $y_1' = c y_3 \\ y_2' = c y_4 \\ y_3' = -f(y_1, y_2)y_2 \\ y_4' = f(y_1, y_2)y_1 $ given $ y_3(-1) = 0 = y_3(1) \\ ...
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68 views

Absorbing BC's / PML on a graph

The wave equation, $$\ddot{u} = c^2 \Delta u,$$ can be generalized to abstract graphs by using the negative graph Laplacian in place of the physical Laplacian. Is there a graph-theoretic analog of ...
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1answer
3k views

Implement Robin Boundary Condition

This is a follow up to this question. I have a nonlinear BVP on $x=0$ to $L$: $$ (T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0 $$ to which I apply a ...
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2answers
3k views

Neumann Boundary Condition at r=0 in Polar Coordinates (Numerical BCs)

I have asked a question in this regard earlier. I am trying to solve the following equation in Polar Co-ordinates: $$ u_t - (u_{rr} + \frac{1}{r} * u_r + \frac{1}{\theta} * u_{\theta\theta} + bu) = ...
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0answers
600 views

Finite Difference in Polar Co-ordinates

Background Please note that I am duplicating the question on scicomp. I have already asked this in math. I am trying to come up with a scheme in Polar Co-ordinates for the following PDE: PDE I am ...
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0answers
377 views

infinite but non-periodic space with PMLs in Comsol [closed]

I have a problem with implementing proper boundary conditions on sides of my simulation (in RF or WaveOptics module of Comsol). I want to obtain an infinite but not periodic space (beacuse then I will ...
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1answer
194 views

Neumann boundary problem

I'm writing a solver for a differential equation with two neumann boundaries (u'(0)=u'(1)=0) and I can't figure out how to determine how to solve the problem. What will my boundaries be and how do I ...
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1answer
287 views

What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as $$ \left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( \begin{array}{ccc}...
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2answers
361 views

Transparent boundary conditions for finite element simulation of TDSE

I have implemented a version of Visscher's method for numerically solving the TDSE (A fast explicit algorithm for the time-dependent Schrödinger equation) (also described in Are there simple ways to ...
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2answers
1k views

Do I need to impose boundary conditions in the Jacobian matrix?

In the framework of Finite Element Method, when the Newton method is used, we solve $J(x^k) \delta x = -f(x^k)$, and the increment $\delta x$ would not change some entries from $x^k$ related to ...
3
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1answer
841 views

How to know whether a boundary-value ODE problem is well defined?

I am using bvp4c from Matlab to solve a boundary values ODEs problem. Given the ODEs and boundary conditions, is there any way to have more information on the solutions? How many do I expect? 0, 1, ...
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1answer
330 views

How to impose an integral conservation in solving ODEs boundary value problems (BVP)?

I have a system of coupled ODEs that I want to solve. The functions are A(x), B(x), C(x). It is a boundary values problem. I am using Matlab bvp4c. So far I am not satisfied with my solutions. For ...
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2answers
855 views

Get symmetric Finite Difference matrix in non Laplacian settings

I would like to solve a system of differential equations $u+\nabla(\nabla\cdot u)=f$ or in more detail $a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$ $b+\partial_x^2b+\partial_x\...
3
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1answer
306 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + f(x,t)...
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0answers
288 views

Treatment of Neumann (Traction) boundary conditions using projection methods

I am looking to solve the incompressible Navier-Stokes equations in 3D, using an inflow boundary condition specifying a velocity: $\mathbf{u} = \mathbf{g}_0 \,\, \forall \,\, \mathbf{x} \in \Gamma_u$ ...
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2answers
697 views

Boundary conditions Chebyshev differentiation

I was wondering if anyone has any experience dealing with boundaries when implementing chebyshev differentiation. I am currently trying to implement a no slip boundary condition to solve the ...
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1answer
212 views

jump conditions for Poisson/Darcy equation in primal form versus mixed form

Consider the Darcy equation, $$\mathbf{v} + \dfrac{k}{\mu_0}\nabla p = \mathbf{f} \\ \mathrm{div}\; \mathbf{v} = 0$$ If the coefficient $k$ is piecewise constant across an interface $\Gamma$ in the ...
3
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1answer
535 views

Jump condition for elliptic equation in standard finite element method

Consider the elliptic PDE $ -\mathrm{div}(k(x)\nabla u) = f(x) \in \Omega$ and $u = 0$ on $\partial \Omega$ If $k(x)$ is piecewise constant across an interface $\Gamma$ in the domain, then we ...
5
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1answer
278 views

Is there a jump condition for this PDE? ( Brinkman model , piecewise constant permeability)

The Brinkman equations for steady flow of an incompressible fluid through rigid porous solid are: $-\dfrac{\mu_0}{k}\mathbf{v} - \mathrm{grad}p + \mu_0 \Delta \mathbf{v} =0$ and $\mathrm{div}(\mathbf{...
5
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1answer
582 views

Applying Dirichlet b.c. to the Eigenvalue-Problem

If you use a FEM (on the variational formulation), you can discretize some continuous eigenvalue problem, $$L u = \lambda u \ \ \text{on} \ \Omega,$$ into some discrete, generalized eigenvalue problem,...
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0answers
2k views

Finite differences and Neumann boundary conditions

I am dealing with a highly nonlinear system of two PDEs. I already have a code to solve the system in case of Dirichlet boundary conditions. The explicit system is: $$ \begin{eqnarray*} \partial_{t}u ...
5
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3answers
4k views

Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
5
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2answers
939 views

Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

I would like to numerically solve a hyperbolic PDE of the form $\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...
2
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1answer
641 views

Method of lines for inhomogeneous Dirichlet conditions

I understand how to set up the boundary conditions for a steady state problem discretized by Galerkin method, for a time dependent PDE below, $$\frac{\partial}{\partial t} u = c\nabla^2 u + a\nabla u ...
5
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0answers
166 views

Elliptical problem with Robin BC

Working in finite differences, I am using a transformation on the temperature variable $\Theta = \int_{T0}^T \kappa(T)dT$ to linearize the steady-state heat equation into a Poisson equation $-\...
3
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3answers
927 views

Solving Stokes flow with walls using Oseen tensor

Introduction I've developed a code to solve for generalised, incompressible 2D Stokes flow $\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$ $\nabla . \mathbf{v} = 0$ where $\mathbf{S}$ can ...
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1answer
233 views

GMRES: Making the matrix square without solving for boundaries

How do we define the matrix for GMRES, if we do not want to solve the boundary elements but only the interior ones. I am using pentagonal elements so in a row there are 6 elements (cell itself + 5 ...
2
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0answers
185 views

boundary conditions with non-constant coefficients in cell centred finite volume method

Suppose am solving the heat conduction equation in 1d with Dirichlet boundary conditions. The thermal conductivity $k$ is a non constant function. So $-(k(x)u'(x))' = f(x)$ The value of $k$ enters the ...
2
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1answer
231 views

Neumann / natural BCs in FEA

I'm trying to work out an as-general-as-possible 2D Laplace example for Finite Element Analysis. Starting from $\Delta u = 0$ for an unknown $u(x,y)$, I multiply both sides (well, in practice only the ...
2
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1answer
435 views

Moving airfoil boundary conditions

I am trying to simulate a moving airfoil with constant speed (Mach=0.755, aoa=1.25). I solve Euler equations with Roe's method. I have two boundary conditions: Farfield Slip wall (airfoil) For all ...
3
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1answer
151 views

Imposing symmetry plane boundary condtition

I want to impose symmetry plane boundary condition for a solid mechanics problem. I googled around and found out that in many places people say to "forbid displacemnts out of symmentry plane and ...
5
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0answers
88 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 ...
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1answer
794 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
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0answers
55 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 \frac{\partial}{\partial\mathbf{...
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0answers
25 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 ...
4
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1answer
618 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: $$ \nabla\cdot(\mathbf{b}...
4
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2answers
240 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
360 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 ...

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