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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42 views

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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40 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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92 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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145 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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37 views

How to avoid density getting “deleted” in two way rigid body coupling with LBM CFD?

I've been reading this paper recently, which talks about using Lattice Boltzmann methods and two way coupling. Specifically, it outlines fluid solid coupling, and solid fluid coupling, and how simply ...
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74 views

Solving a system of PDEs with no-flux boundary conditions (finite difference discretization)

I am interested in solving a system of linear PDEs with the finite difference method and I'm having trouble to solve the no-flux boundary condition correctly. \begin{align} \frac{\partial n}{\partial ...
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72 views

Solving diffusion equation using finite difference method

I am solving an 1-dimensional diffusion equation with Neumann boundary condition at outlet and constant concentration, C, at the inlet. In the end, I want to observe how the concentration diffuses ...
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108 views

FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\...
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38 views

Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme

I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$). $\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$ Where $...
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114 views

PDE discretization on triangular domain

Given the 2D Poisson equation $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$ defined on the domain $\Omega := \{(x,y) \in \...
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42 views

Hydrogen-like wavefunction as starting guess for atomic solver?

I've been looking into radial solvers for quantum wave equations (Schroedinger and Dirac). In both cases, the suggestion seems to be to go with the "shooting method", with integration schemes of ...
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81 views

Implementing boundary condition

I'm studying the transport of species A in the blood vessels, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ At x=0, I want to use the ...
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343 views

Inflow and outflow boundary conditions for advection-diffusion equation

I'm trying to solve this advection-diffusion equation (ADE): $$\frac{\partial \phi}{\partial t} + \nabla \cdot (-D \nabla \phi + \mathbf{u} \phi) = 0$$ In fact, this ADE framework is coupled to a ...
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52 views

Decrease in slope during convergence analysis

I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides. I used 5 refinements: $dx = dy = dz = ...
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123 views

Method of manufactured solutions - choice of type of boundary conditions

I am attempting to use the method of manufactured solutions (MMS) for code verification for linear elasticity. However, this is more of a general question regarding the general use of MMS. In MMS, ...
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216 views

Neumann boundary condition FD implementation for instationnary diffusion equation

I am trying to solve this diffusion equation : $\dfrac{\partial D\dfrac{\partial f}{\partial x}}{\partial x}+S = \dfrac{\partial f}{\partial t}$ ($D$ is not constant and varies according to $x$) with ...
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70 views

Radially symmetric system of PDEs in deal.II

I am trying to solve the radially symmetric polar form of the PDE with homogeneous Neumann BC in deal.II on a unit circle: $$ u_t = \Delta u - \nabla \cdot (u \nabla h) $$ $$ h_t = \Delta h $$ I am ...
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398 views

Structural mechanics traction boundary condition question

In structural mechanics, are the boundary conditions "free surface," "Traction free", "stress free" all equivalent Neumann boundary conditions?
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238 views

the augmented global stiffness matrix is not positive semi-definite using Lagrange Multipliers method within FEM

The augmented global stiffness matrix is not positive semi-definite when using Lagrange Multipliers method to enforce boundary constraints on a simple square domain of integral form: I am considering ...
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53 views

Appropriately handling boundary conditions in a PDE eigenvalue problem

Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically,...
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160 views

shallow water equation maccormack method

I am trying to make a code for 1D shallow water equation (nonlinear without source terms) using the MacCormack method for sinusoidal wave propagation. My issue is that the wave fluctuates and does not ...
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53 views

Extrapolating to non-fluid cells (for Shallow Water Equations), for a shore/beach?

The water height $h$ and 2d velocity field $(u,w)$ are "extrapolated to non-fluid-cells, i.e., setting $h$ equal to the value in the nearest fluid cell." [Bridson] I'm using finite differences. ...
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59 views

Discrepancy in estimating boundary stencil for finite difference method

I am trying to estimate the FD stencil for boundary as mentioned in this paper (section 4.1) using MATLAB. The stencil order (6th) is higher than the one mentioned in paper (4th). $$ f_1' +\alpha f_2'...
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112 views

How would you specify mixed boundary conditions for a 2D PDE in the matrix used for finite differences

I have the following PDE in 2D: $U_{x} + U_{xx} + U_y + U_{yy} + U_{xy} = f$ where $f$ is a constant. And I'm trying to create a matrix $A$ to solve the PDE through finite differences: $AU = f$. I ...
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39 views

Simulating periodic boundary condition with long_range interaction

Consider 2 particles in a 2D plane. There is a long-range interaction between two particles: $$f (r_i-r_j)= \frac{\vec{r_i} - \vec{r_j}}{|r_i-r_j|^3}$$ $$\vec{r_i} =(x_i,y_i)$$ $r_i$ determines the ...
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75 views

Numerically solving generalized eigenproblem with Neumann conditions

I am interested in finding the eigenvalues/eigenfunctions of problems such as $$ \partial_{xx} u = \lambda \partial_{yy} u, $$ which can be solved as the generalised eigenvalue problem $$ \mathbf{A}...
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572 views

Implementing Neumann boundary condition for elasticity problem using the finite element method

I am solving the following mixed boundary elasticity problem with the finite element method on the unit circle, using a triangle mesh with 3 nodes on each element. In my problem I know $u_1$ and $t_2$...
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385 views

Implement Robin boundary condition (finite volume)

I have a PDE equation with Robin Boundary condition in an annulus system and I should solve it by finite volume method: \begin{align} \frac{\partial T_f}{\partial t} - k \left(\frac{\...
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599 views

How to implement outgoing wave boundary condition

I am solving the one dimensional wave equation: $0=\Box\phi = -\partial_t^2\phi + \partial_r^2\phi ,$ using a Crank-Nicolson finite difference scheme, in the domain $r\in[0,R]$. First, I define $\xi\...
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99 views

Discontinuos Galerkin Method - inhomogeneous advection problem

I'm currently trying to get into this topic. I've learned that the basic scheme for the advection problem ($D_{x}u+a*D_{t}u=0$) can be solved in a scheme like $$ M^{k}\frac{d}{dt}u^{k}_{h}-(S^{k})^{T}...
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127 views

boundary condition in 2-D planar finite difference problem

I'm working on a 2-D planar finite difference code. My differencing scheme at the boundary nodes involves introducing ghost nodes in the computation. My code also involves a multi-dimensional ...
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258 views

Breather solutions of Sine-Gordon Using Finite Differences

I'm attempting to simulate a standing breather of the form $$ u(x,t)=4\tan^{-1}\left(\sqrt{3}\cos\left(\frac{t}{2}\right)sech\left(\frac{\sqrt{3}x}{2}\right)\right)$$ for the Sine-Gordon equation $$u_{...
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200 views

Spherical Advection Discretization (boundary nodes)

Consider the spherical advection problem: describing the conservation of a property $u$ in a closed spherical domain. $$ \frac{\partial u}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^...
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327 views

How to add a Ricker Wavelet (Mexican Hat) to a 2D/ 3D fem mesh?

I have a 2D square mesh and a 3D beam shaped mesh and I want to propagate a seismic wave in them. I am trying to simulate them using Open source FEM codes (fenics). I have left the top surface to be ...
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589 views

Issues with self-consistent Poisson-Schrodinger solver

I'm currently in the process of writing a self-consistent Schrodinger-Poisson solver for a device heterstructure (High Mobility Electron Transistor). The algorithm is based off of this journal(1). I ...
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451 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 ...
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142 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 ...
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92 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 ...
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54 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 \end{array}\...
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84 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 guess)...
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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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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 ...
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1answer
77 views

deal.ii - ParaView “warp by scalar” of my output is not continuous

During our finite element course, we've solved the linear elasticity problem in 2D on a square (GridGenerator::hyper_cube) with $Q_1$ bilinear finite elements in ...
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23 views

Reynolds boundary conditions

I came across this paper comparing various boundary conditions. I am particularly interested to understand how to obtain the Reynolds boundary conditions (refer to equation 28). $$\left( \frac{1}{c} \...
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35 views

Insert a boundary condition without removing periodicity assumption

I've to perform a multi-fluid internal flow simulation with a code which intrinsically assumes periodicity on a given direction (say z) on a cartesian grid. Nonetheless, the problem I'm trying to ...
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85 views

Boundary conditions for an FEM approximation of the Laplace operator

Using FEM, I want to approximate the Laplacian $$u = \nabla \cdot \nabla h \, ,$$ where $h(x,y)$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable. I am using MOOSE ...
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24 views

(FD WENO) Correct symmetry boundary condition for Euler equations

I'm trying to solve 2D Euler equations in axisymmetric formulation with finite-difference WENO scheme. I found some info on high-order boundary conditions for plane formulation (in this thesis, for ...
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120 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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46 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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1answer
92 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 ...