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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1answer
60 views

What are acceptable boundary conditions for porous media flow?

I am attempting to simulate fluid flow through a porous foam. I would like to have no-slip boundary conditions on part of the boundary and free flow conditions on the inlet and outlet. Right now I am ...
1
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1answer
91 views

Please explain the meaning of these Boundary conditions [closed]

I am trying to learn Gmsh and Fenics and was looking at an example which shows the application of Boundary conditions on a simple Poisson problem. Here is the link: ...
4
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1answer
122 views

What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
0
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0answers
32 views

Compute solution of a pde with multiple boundary conditions

What are some general methods which can allow to solve the equation $-\Delta u = 0$ on a two dimensional domain, with mixed boundary conditions? There are a few methods I have in mind: finite ...
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1answer
61 views

Solving a linear equation system with pure Neumann condition

I am trying to solve a linear equation system $\textbf{A}\textbf{x}=\textbf{b}$, e.g. a Poisson equation discretized in strong form, using biCGstab method. Since there are only natural Neumann ...
2
votes
1answer
37 views

Finding the frequencies of vibration of a circular and square drum

I want to find the frequencies of vibration of a circular and square drum. To do this, I need to solve a 2-dimensional wave equation (PDE) with boundary conditions. Every method that I have researched ...
0
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1answer
39 views

oscillating flow inlet BC

I want to do LES of an oscillating flow i.e. a sinusoidal flow without a mean component in a channel. In order to have a fully developed flow at lower mesh count without using a long channel I want to ...
1
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1answer
47 views

Satisfying Periodic Boundary Conditions while plotting spherical particles inside a cube

I am trying to plot spherical particles in a cube of fixed dimension in matlab. I face a problem here where the center of the sphere is too close to the edge of the cube in this case the rest of the ...
0
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1answer
71 views

Simulation of Laplace Equation in 3-D with mixed BC of Dirichlet-Neumann

I am simulating a Mixed-Boundary value (Dirichlet-Neumann) problem using Finite Difference Method on a unit 3-D cube such that the left, lower, and front plane have $u=u(x,y,z)=1$ (Dirichlet) and ...
3
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2answers
112 views

Is “tangent stiffness matrix” the same as “stiffness matrix”?

I'm trying to implement nonzero Displacement Boundary Conditions in VegaFEM on a non-linear model, using the method outlined in §3.6.2 of University of Colorado's intro to FEM (modify $f = Ku$: set ...
3
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0answers
60 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
99 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
votes
1answer
144 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 ...
0
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1answer
119 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 $N$ ...
0
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1answer
29 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
60 views

Solution to PDE with differential boundary conditions

I have the following equations $$ a_t(x,t)=1-a(x,t)b(x,t)^\gamma+D_1a_{xx}(x,t) $$ and $$ b_t(x,t)=\alpha(a(x,t)b(x,t)^\gamma -b(x,t))+D_2b(x,t)_xx $$ where $a,b:]0;4\pi[\times \mathrm{R}_+ ...
2
votes
0answers
102 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 ...
1
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2answers
169 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} &- ...
1
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1answer
79 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 ...
3
votes
1answer
83 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 ...
2
votes
2answers
226 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 ...
0
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1answer
168 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 ...
2
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0answers
88 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 ...
1
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1answer
61 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) \\ ...
3
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0answers
52 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 ...
0
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1answer
119 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 ...
3
votes
2answers
259 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) = ...
0
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0answers
91 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
43 views

infinite but non-periodic space with PMLs in Comsol

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 ...
0
votes
1answer
91 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 ...
1
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1answer
172 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( ...
1
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1answer
53 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 ...
3
votes
2answers
204 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 ...
2
votes
1answer
219 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, ...
1
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1answer
88 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 ...
3
votes
1answer
145 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 $a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$ ...
3
votes
1answer
128 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 + ...
2
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0answers
91 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: $\mathbf{u} = \mathbf{g}_0 \,\, \forall \,\, \mathbf{x} \in \Gamma_u$ ...
2
votes
1answer
154 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 ...
0
votes
1answer
90 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 ...
2
votes
1answer
124 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
votes
1answer
191 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 ...
4
votes
1answer
119 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 ...
0
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0answers
246 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 ...
4
votes
3answers
608 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
votes
2answers
234 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
votes
1answer
153 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 ...
4
votes
0answers
67 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 ...
2
votes
3answers
253 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 ...
0
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1answer
83 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 ...