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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60 views

How are finite volume method boundary conditions implemented without using ghost-cells?

I'm currently trying to implement my own FVM code in cpp, but when I try to calculate the laplacian of a test function, given by \begin{align}\phi_0=\sin(2\pi x)\sin(2\pi y)\end{align}, I get ...
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64 views

Poisson equation, stiffness matrix positive definiteness, Dirichlet boundary conditions

I have a question regarding the positive definiteness of the stiffness matrix. Specifically, I believe that it should be positive definite only when at least one Dirichlet point is given, so I would ...
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0answers
39 views

How to solve this boundary value problem which has more unknown than equation on MATLAB

I need your helps about solving the problem below with MATLAB. I am trying to solve 2D Stress Wave Propagation problem by using FDTD(Finite difference time domain) method on the cylindrical coord. I ...
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34 views

Non uniform grid for divided differences

Construct a difference method for the the numerical approximation of the solution of the following boundary value problem:Find $u:[a,b]\to \mathbb{R}$ function,such that $$ -u''(x)=f(x)$$ and $u(a)=0, ...
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41 views

Cauchy problem ill-posed?

Find the solution to the Cauchy problem consisting of the wave equation : $$u_{xx}-u_{yy}=0$$ together with initial conditions: $$ u(x,0)=0,$$ $$u_{y}(x,0)=g(x)$$ for some known initial datum $g$. Is ...
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2answers
92 views

Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?

I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions. So far, I have found the problem solved analytically in one ...
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1answer
56 views

Implementing Dirichlet BC for the Advection-Diffusion equation using a second-order Upwind Scheme finite difference discretization

i am implementing a Matlab code to solve the following equation numerically : $$ (\frac{\partial c}{\partial t} =-D_{e} \frac{\partial^2 c}{\partial z^2} +U_{z}\frac{\partial c}{\partial z}) $$ with ...
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39 views

“This DAE appears to be of index greater than 1” daeic12 (line76) error code

Hi I am trying to solve a set of pde converted into ODE and DAE using central finite difference method. I have used the MATLAB 'solve' command to determine the coefficients of fictitious nodes for ...
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39 views

Integrating a wavelike equation with absorbing boundary conditions

I am trying to numerically solve the following equation: $\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with: $\phi(x, 0) = I(x)$ ...
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27 views

Electric Current Module in Comsol

I am trying to solve for the potential in the electric current module in Comsol. I have a wire with 2 current boundary conditions where the in equals the out. The model can be solved in stationary but ...
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A Question About a Claim from 1991 Computational EM paper about the Cancellation of certain Boundary Terms

Please let me know if this is not the appropriate site for this question. I found questions regarding EFIE/MFIE/CFIE on this site, so I thought my question might fit. I am studying the paper by Putnam ...
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2answers
67 views

inclined/general Dirichlet boundary conditions

For simpilcity, consider a single quad linear elasticity finite element in 2D. The Dirichlet boundary conditions on node 1 and node 2 are easy to implement and can be handled in the standard way. ...
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1answer
55 views

Are spatial boundary conditions required for PDEs discretized with Method of Lines?

As far as I understand, you need to define boundary conditions in time and space to select a unique solution to a PDE and make it solvable. However, in ODEs I only need to specific the initial value ...
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1answer
59 views

Where could error terms that blow up in SWE come from?

I have been working on a solver for shallow water equations with reflective boundary conditions. I have found that it diverges very fast. As a workaround I noticed yesterday that if I smooth the ...
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64 views

How to use FEniCS to calculate the electric field of an isolated charged sphere

Initially I thought that this is the kind of question which ought to have already been answered in the form of an example online, but so far I haven't found one. I will admit that I am very new to ...
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1answer
103 views

How to apply Dirichlet boundary conditions to time-dependent PDEs?

Assume the time-dependent linear elasticity equation. Using a finite element discretization we obtain $$M\ddot{u}=Ku+F_\text{ext}$$ where $M$ is the mass matrix,$K$ is the stiffness matrix, and $F_\...
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1answer
158 views

Finite element (1D) for steady state non-linear problem

I need to solve with linear finite elements the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ \sqrt{u} \frac{\partial u}{\partial x} \Bigr] =0$...
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2answers
181 views

No flux Neumann boundary condition for non-stationary PDE equivalent to Dirichlet boundary?

When using no flux Neumann boundary conditions (i.e. zero derivative to the normal on the boundary) in a non-stationary PDE, I don't seem to recognize the difference to using Dirichlet boundary ...
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1answer
142 views

Imposing no flux boundary conditions on variants of the Cahn-Hilliard equation using Finite Differences in Python

I have been looking into simulations of phase separation in variants of the Cahn-Hilliard system and have been running into issues with implementing no flux boundary conditions on certain variants. ...
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33 views

Numerical simulation for a bounded process. Is slight deviation a “normal” fact?

Suppose I have to numerically simulate a process $\{y_t\}$ such that $y_t\geq0$ $\forall t\in\mathbb{N}$, with $t$ denoting time-step. Let's suppose I use MonteCarlo with $\mathscr{N}$ simulation ...
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37 views

Discretization formula for a system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
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1answer
95 views

traction boundary conditions in elasticity

I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My ...
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1answer
97 views

Which scheme for inhomogeneous convection-diffusion equation with highly variable coefficients?

I have a 1D convection-diffusion equation $\sigma_t = a(x,t) \sigma_{xx}+b(x,t)\sigma_x+f(x,t)$ defined on the unit interval, with nonzero Neumann boundary conditions at both ends. It should be noted ...
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Euler Explict Method for solving the PDE for option prices under the Schwartz mean reverting model. Numerical Finance

I have to solve the following PDE for a Call option : $\partial_tV + \{ \alpha - (\mu - \lambda/ \alpha -log(S))\}S\partial_SV + 1/2 \sigma^{2}S^{2}\partial_{S}^{2}V - rV = 0$ Where V(S,t) is the ...
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15 views

A way to generate unit lattices from a honeycomb structure

I am looking to make certain computations on the vertices of periodic cubic honeycombs and quasiregular honeycombs like tetrahedral-octahedral honeycomb. Cubic are simple enough and amount to generate ...
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2answers
223 views

Finite difference method having a discontinuity

I am trying to understand the FDM which is a widely used method solving differential equations by using approximation below. $$\dfrac{\partial u}{\partial x}=\dfrac{u(i+1)-u(i-1)}{2\Delta x}$$ How can ...
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0answers
38 views

Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
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0answers
25 views

At what l/d ratio will a frame element start to behave as a shell element?

I'm working in ETABS. There are few columns of dimension 300mmx1400mm. The height of building is 36.6 meter above ground level and the dimension of building is 26mx68m. I'm getting the time period of ...
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1answer
109 views

Methodology Suggestion for Wave-propagation Problem using Finite Elements

I want to simulate the propagation of a sinusoidal plane wave in a rectangular domain using Finite Elements Method. First, the wave should propagate through a fluid medium, then it will encounter a ...
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43 views

COMSOL Circularl polarization

I'm having some problems trying to implement circularly polarized light in COMSOL Muliphysics. For a isotropic homogenous media, I've obtained without problems the TE and TM reflectance curves. ...
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2answers
302 views

Kronecker product representation of the finite difference laplacian

The laplacian equation when discretized gives a system of linear equations that can then be solved. See the answer to this question: https://math.stackexchange.com/questions/3120948/discretization-...
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17 views

Shallow water equations: boundary conditions for sub- and super-critical flow

This question is (sort of) a continuation of this previously asked question. I am wondering about, in general, how we construct well-posed boundary conditions (both continous and numerical) for flow ...
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1answer
352 views

How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
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74 views

Open Boundary Conditions for Solving Navier Stokes in moving ALE domain

I'm trying to solve the problem of a body freely rising in a fluid due to gravity, the density of the body is just slightly less than that of the fluid. The body is rigid, so I have to solve a ...
2
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1answer
178 views

Boundary conditions in a finite element eigenvalue problem

I've been reading multiple papers and related posts for a while now, but I can't seem to find a specific answer to the issues I'm having so I hope someone can clarify things here. I'll provide some ...
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0answers
71 views

Applying boundary Conditions on FEM

I have a partial differential equatons as shown below. $$\dfrac{d}{dx}((1+x)\dfrac{du(x)}{dx})=0$$ With the following boundary conditions. $$u(0)=0, u(3)=10$$ To solve it using FEM, I multiplied the ...
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1answer
93 views

Applying weak form

I have two dimensional equation and I want to solve it using Finite Element Methods. $$ \nabla . (\alpha(x,y)\nabla u(x,y)) + \dfrac{\partial u(x,y)}{\partial x}+\dfrac{\partial u(x,y)}{\partial y}+u(...
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12 views

Differential parameterized inequalities

Let $H$ be an Hamiltonian and denote $\vec{H}$ the associated Hamiltonian vector field. I am interested in solving numerically the following problem $$ \dot z(t) = \vec{H}(z(s),t_1,\ldots, t_p) \...
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2answers
151 views

Solving a Boundary Value Question $\frac{d^2y}{dx^2}=y\cos(x)+\frac{\sin(x)}{x^2+2}$ using Python

I'm looking to solve this boundary value question using the shooting method! $$\frac{d^2y}{dx^2}=y\cos(x)+\frac{\sin(x)}{x^2+2}$$ given the initial values: $$y'(x=-1)=-1\\y'(x=5)=0$$ I'm aware of ...
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1answer
76 views

Determining Displacement Field on a Sphere

I need to find the displacement field for this sphere shape in terms of $\delta$. So far, by applying boundary conditions, I know $u_r = u_\theta = u_\phi = 0$ at $r = a$ and $u_r = \delta, u_\theta = ...
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0answers
36 views

Boundary Conditions for Continuum Mechanics

If I'm given some sort of shape and there is a displacement given but there are no external forces acting on the body, do I still need to write down the traction free boundary conditions or will there ...
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1answer
70 views

Simulating pressure waves at an impedance boundary

I am trying to simulate pressure waves crossing a boundary from one medium to another (e.g., water to air) in Matlab. The code that I have got so far, which is largely taken from Wikipedia on Partial ...
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0answers
82 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
78 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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0answers
100 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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0answers
83 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
145 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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0answers
48 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
59 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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0answers
40 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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