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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-1
votes
0answers
43 views

what is procedure for crank Nicolson method in nonlinear partial differential equations? [closed]

can u tell me step by step of what is procedure for crank Nicolson method to apply in nonlinear partial differential equations? and how to plot it.
3
votes
2answers
155 views

Determine numerical infinity for Schrodinger equation $−\psi''(x) + x^ 2 \psi(x) = E\psi(x)$

Consider the following Schrodinger equation for the harmonic oscillator with real $x$: $$ −ψ''(x) + x^ 2 ψ(x) = Eψ(x). $$ I solve the last equation using shooting method and implicit Runge-Kutta ...
3
votes
1answer
30 views

Apply second order finite difference discretization for mixed boundary condition

I want to solve the problem below \begin{equation} \begin{aligned} \eta u-\Delta u &=f, &\text{in $\Omega$}\\ \end{aligned} \end{equation} where $\Omega=(0,1)\times ...
1
vote
1answer
94 views

Outflow boundary condition

I know that in outflow boundary we assume a zero normal gradient condition and use upwind scheme for approximation. However, I saw this sentence in a book which I do not understand; "Convective fluxes ...
0
votes
1answer
57 views

Boundary conditions generalized eigenvalue problem

Consider the following eigenvalue problem \begin{equation} \mathcal {L} x(s) = \lambda x(s), \end{equation} where \begin{equation} \mathcal {L} = \alpha \partial^4_s + (s^2-1)\partial^2_s + s ...
2
votes
1answer
115 views

How do I program periodic boundary conditions? [duplicate]

Hi I have a code below that solves non linear coupled PDE's given Dirichlet boundary conditions. However I need to implement periodic boundary conditions. The periodic boundary conditions are ...
5
votes
1answer
76 views

How can i convert a boundary flux condition into an internal source term?

Imagine a rectangular box that is thermally insulated on all sides except for one, where a heat flux is applied. Now imagine the same box with the same conditions on all sides except that the side ...
1
vote
1answer
67 views

Finite difference scheme for Webster equation

Webster equation is a popular generalization of 1D wave equation used for ducts of variable cross-section $S \equiv S(x)$. Assuming harmonicity in time, the spatial equation for propagation of ...
0
votes
0answers
37 views

Using Periodic BC for Taylor Green Vortex Flow Simulation

I want to solve the very simple case of two dimensional Taylor Green Vortex Flow. I am using incompressible Navier Stokes equations and Artificial Compressibility Method as my solver where I introduce ...
0
votes
0answers
46 views

Periodic boundary conditions for solving Navier Stokes Equations on a Staggered Grid

I want to solve two dimensional Navier Stokes equations on a staggered grid for the case of Taylor-Green Vortex. My initial conditions are standard sine and cosine functions. As I am aware, I should ...
3
votes
0answers
44 views

Reflecting boundary condition posed as a Riemann problem

I am trying to implement a solver for the Euler/Navier Stokes equations. I have a problem implementing boundary conditions for the wall. I am using an unstructured solver. A lot of literature says ...
1
vote
1answer
73 views

How to deal with PDE over the real line

I have a PDE defined over $\mathbb{R}$, for which I don't have the exact solution, and I am to approximate it with finite differences so I need to input some BC. Can anyone suggest any good ...
1
vote
1answer
145 views

Shooting method - Matlab ODE

I'm trying to solve these equations of hypersonic adiabatic flow over a flat plate. I did all the simplifications and got these equations for the stagnation point flow. $$\left(Cf''\right)' + f f'' = ...
0
votes
1answer
74 views

boundary conditions of linear advection problem

I am solving the 1D advection problem given by: $$\frac{\partial u}{\partial t}=-c\frac{\partial u}{\partial x}$$ where c is the wave speed, and u is the unknown field variable, and x and t are time ...
6
votes
1answer
145 views

Poisson equation finite-difference with pure Neumann boundary conditions

I'm trying to solve a 1D Poisson equation with pure Neumann boundary conditions. I've found many discussions of this problem, e.g. 1) Poisson equation with Neumann boundary conditions 2) Writing the ...
1
vote
1answer
121 views

Numerical method for a BVP with mixed boundary conditions (MATLAB)

I've been given a second-order non-linear ODE: $$\frac{d^{2}\theta(s)}{ds^{2}} = sf_{g}\cos{\theta} + sf_{x}\cos{\phi}\sin{\theta}$$ where $f_{g}, f_{x}$ and $\phi$ are constants. The boundary ...
1
vote
0answers
81 views

Boundary Conditions for the given PDE

I'm working on the Black-Scholes equation, but I'm pretty new to financial modeling. Right now, I am trying to understand the Black-Scholes PDE. I understand that the Black-Scholes equation is given ...
1
vote
0answers
54 views

Center of mass in systems with periodic boundary conditions

I have a question about the calculation of center of mass (COM) in systems with periodic boundary conditions. There is a method introduced here: ...
0
votes
0answers
30 views

Boundary conditions shooting method

I am trying to solve the differential equation $\frac{d^{2}y}{dr^2}+(\frac{1}{r}+1)y=0$ with the boundary conditions $y(r) \rightarrow r \frac{dy}{dr}(0)$ as $r \rightarrow 0$ and $y(r) \rightarrow ...
0
votes
0answers
53 views

problem with understanding the fluid boundary conditions of a 1D probelm

I am having problems understanding the boundary conditions of the problem described in this paper on researchgate Essentially the problem consists of a one dimensional fluid chamber in contact with a ...
1
vote
1answer
131 views

BCs in a coupled problem

Consider a thermo-mechanical coupled problem, where coupling exists from both the sides, mechanical loading producing thermal effects and vice versa. In such a case, is it necessary to always ...
3
votes
2answers
137 views

Understanding Neumann BC

I understand what Neumann BC means physically and how to imply them. However, I am not able to perfectly understand it the way it is represented mathematically as $\partial u / \partial n$ where $n$ ...
0
votes
1answer
128 views

How to implement 'stress-free' boundary conditions (stokes flow)?

I have to implement 'stress-free' boundary conditions for a stokes flow of a 2D-domain (for the left, bottom and right boundaries). The boundary condition are also described mathematically as $2 \eta ...
4
votes
1answer
96 views

How to physically understand time dependent boundary conditions?

I am a beginner in Computational science and FEM. I came across some PDEs which implement time dependent boundary conditions. I am not able to visualize exactly a physical scenario of how that would ...
2
votes
1answer
80 views

Laplace's equation with periodic Dirichlet boundary conditions

Consider a Laplace's equation with Dirichlet boundary condition: ${\nabla ^2}\Phi = 0$ in a domain $D$ with given Dirichlet Boundary condition: $\Phi=\Phi_o$ at $\partial D$ (smooth, but not ...
2
votes
0answers
77 views

Finite difference aproximation - Darcy law

I am solving following problem: Filtration of water can be described in bi-dimensional case by $$- \partial_x(K(x,y) \partial_x u ) - \partial_y (K(x,y) \partial_y u ) = 0, $$ where $u$ - water ...
2
votes
0answers
95 views

Solving constrained BVP, singular Jacobian

The boundary value problem is $$ \begin{cases} \dot{x}_i = \begin{cases} (0.5D^{-1}\psi)_i, \text{ if }(0.5D^{-1}\psi)_i \le 0 \\ 0 \text{, otherwise} \end{cases} \\ \dot{\psi} = 2\Sigma x \\ x(0) ...
4
votes
0answers
140 views

Boundary conditions in the Finite Element Method at only one side of computational domain

I want to solve a Sturm-Liouville problem in 1D, i.e., \begin{align} [p(x)\ u'(x)]'+q(x)u(x) = f(x) \end{align} with boundary conditions \begin{align} u(0)=a \hspace{1cm} u'(0)=b \end{align} How do I ...
3
votes
1answer
93 views

LU-SGS and boundary conditions

I am trying to understand how boundary conditions are implemented when one uses the nonlinear LU-SGS algorithm for Euler equations. Most papers describe the Gauss-Seidel sweep over mesh cells, but do ...
3
votes
1answer
100 views

Smoothed particle hydrodynamics bounduary (ghost particles) properties

I am learning SPH method. At the moment I am trying to implement simulation described in this very good article. However I don't get how the ghost particles properties are computed: Position of the ...
2
votes
0answers
148 views

FEniCS: both normal and shear stress boundary conditions for elasticity? [closed]

I would like to have both the normal (xx) component and shear (xy) component of a 2D (stress) tensor defined on a boundary (y=const, for instance) for an equation which is of the type $$ \nabla \cdot ...
3
votes
2answers
107 views

How to implement boundary conditions in heat equation with no flux and fixed value at the same time? Is it Robyn BC?

I am modeling the temperature of the groundwater using heat equation. I have Dirichlet BC at the top but at the bottom I have constant temperature equal 12 degrees C (see attached pic). It is look ...
4
votes
2answers
158 views

FEM: Possible to have boundary conditions “inside” the domain?

I work on geological problems and I use the Finite Element Method. But this question can be applied on other classical mechanical problems. I work on implicit 3D surfaces (which represent the limits ...
5
votes
1answer
171 views

Do the class of PDEs that lack initial conditions have a name?

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation. Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
1
vote
2answers
196 views

Dirichlet BCs - alternative implementation methods

I am having problems with solving a hyperbolic wave problem with Dirichlet BCs. I have tried reducing the time step sizes, which does not affect the results, and notices increasing the number of nodes ...
1
vote
0answers
45 views

should boundary conditions be effecting moving mesh results?

I have a question on the use of moving mesh to solve the inviscid euler equations. I have solved the following equations: $$\frac{\partial}{\partial t}\left[\begin{array}{c} \rho\\ \rho u ...
1
vote
0answers
49 views

Jacobi method converging then diverging

I am working to solve Poisson's equation in 2D axisymmetric cylindrical coordinates using the Jacobi method. The $L^2$ norm decreases from $\sim 10^3$ on the first iteration (I have a really bad ...
3
votes
0answers
324 views

What should be the number of boundary conditions of a PDE [closed]

As far as I know, for getting a unique solution to a PDE we should impose some boundary conditions to the PDE. "The number of required auxiliary conditions is determined by the highest order ...
0
votes
0answers
244 views

Finite Difference Method Neumann Boundary Condition with Variable Coefficients by Ghost Points

I have found an alternative solution to the problem stated here. Here's the alternative solution link. It says that "A common technique in implementations of $\partial u/\partial x=0$ boundary ...
2
votes
0answers
99 views

Implementation of no-slip boundary conditions in lattice Boltzmann method fluid simulation

My faculty advisor recommended that I take a look at the lattice Boltzmann method as an introduction to scientific computing and potentially an undergraduate honors thesis topic. I cooked up a some ...
3
votes
2answers
215 views

How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices

I am wondering how Dirichlet boundary conditions in global sparse finite element matrices are actually implemented efficiently. For example lets say that our global finite element matrix was: $$K = ...
0
votes
0answers
56 views

Injection Vs Full Restriction in Dirichlet-Neumann 3-D Multigrid

I have implemented the Multigrid method for a Mixed Dirichlet-Neumann boundary value problem where $\nabla^{2}{u}=0$, $u = 1+x+y+z$ for Dirichlet and $\frac{\partial e}{\partial n} = 1$ for Neumann ...
4
votes
2answers
187 views

Absorbing boundary conditions for acoustics in Discontinuous Galerkin

Note: I'm trying to implement a Discontinuous Galerkin method, as kind of a way to learn about these things. As of now, I've taken the acoustic wave equation $c^2 \nabla \cdot \nabla u(x,t) - ...
2
votes
2answers
183 views

Scipy OdeInt solver with Neumann boundary conditions

I'm using scipy.odeint to solve Fisher-Kolmogorov equation: \begin{equation} u_t = u_{xx}+u(1-u) \end{equation} The code can be found here. From Ablowitz and ...
3
votes
2answers
41 views

Boundary conditions for solving Poisson's Equation with Experimental Data

I want to numerically (with Matlab) solve Poisson's equation : $ \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = f(x,y)$ On a rectangular domain using experimental data. From ...
1
vote
1answer
69 views

Periodic boundaries - implementation strategies

I managed to implement the Nearest-Neighboor Ising Model with periodic boundary conditions, it was doable. I also made a modified version of it, where the interaction would go further than the nearest ...
3
votes
2answers
216 views

periodic boundary conditions for triclinic box

I am trying to do analysis on a data set of atomic coordinates generated form lammps. I simulated an alpha glycine crystal in a triclinic box. The box vectors look like the following, where xy,xz,and ...
6
votes
2answers
205 views

Periodic boundary condition for the heat equation in ]0,1[

Let us consider a smooth initial condition and the heat equation in one dimension : $$ \partial_t u = \partial_{xx} u$$ in the open interval $]0,1[$, and let us assume that we want to solve it ...
2
votes
1answer
142 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
vote
1answer
128 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: ...