All Questions
Tagged with finite-difference boundary-conditions
111
questions
1
vote
0
answers
33
views
Imposing higher order finite difference schemes for boundary value problems on a finite interval
I have some questions. I'm going to assume everything is in 1d with a Laplacian operator. If I discretize the Laplacian operator using $p = 2a+1$ grid points with periodic boundary conditions, I ...
- 111
2
votes
1
answer
116
views
Are there necessary conditions for SOR algorithm to converge when used for 2D discrete Poisson problem, with PBC, besides solvability condition?
I am trying to solve a 2D poisson problem that is supposed to represent diffusion of chemicals on a grid: $\nabla^2 R_{ij}=f_{ij}$. I am discretizing the problem with the standard central ...
1
vote
1
answer
62
views
How to calculate the force of solid applied by fluid? Using finite difference method, DNS, staggered grid, SIMPLE algorithm, immersive boundary
Problem
I am using finite difference method to solve classic problem of flow around cylinder, for validation of my group's immersive boundary method.
The common way to validate numerical method is ...
- 31
3
votes
0
answers
84
views
finite difference method not working when going to two dimensions
I have two coupled ordinary differential equations in the steady state:
The following code solves, using the Jacobi finite difference method, in 1d using Dirichlet boundary conditions for function $...
- 131
3
votes
1
answer
277
views
How to evaluate the points near/at the boundary when using Richardson extrapolation for improved accuracy of a derivative
If we want to improve the accuracy of our numerical estimation of a derivative, we can use Richardson extrapolation. The method is very beneficial when using a centered difference scheme and the ...
3
votes
1
answer
209
views
First derivative central differences with reflecting boundary conditions
I have the following problem in 1-D:
\begin{equation}
\partial_t u(t,x) = v(x)\partial_x u(t,x) + \kappa(x) \partial_{xx} u(t,x),\,x\in\Omega;\quad \partial_{\nu}u(t,x) = 0, \,x\in\partial\Omega.
\end{...
- 2,267
0
votes
2
answers
237
views
How and when to impose inhomogeneous Dirichlet boundary conditions in Newton method solver for a PDE?
I am applying a central Finite Difference scheme in space and an implicit Euler scheme in time on a variant of the 2d Burgers equation, of the form:$$u_t + uu_x + uu_y = \nu(u_{xx} + u_{yy})$$ where $...
3
votes
1
answer
769
views
Solving Poisson's Equation with Periodic Boundary Conditions
So, I've been attempting to design a simple solver for a problem of finding the gravitational potential of a system using Poisson's equation (let's call the potential phi, $\phi$). The goal is that I ...
- 33
1
vote
1
answer
366
views
Crank Nicolson Method with closed boundary conditions
I want to simulate 1D diffusion with a constant diffusion coefficient using the Crank-Nicolson method.
$$\frac{\partial u (x,t)}{\partial t} = D \frac{\partial^2 u(x,t)}{\partial x^2}.$$
I take an ...
- 111
1
vote
0
answers
101
views
How to include zero flux boundary conditions?
I am trying to solve the following differential equation in the domain of $\theta \in [0, 2 \pi]$ using finite differences scheme:
For $0< \theta \leq \pi$
\begin{align}
\rho_i^{n+1}=\rho_i^{n}+D\...
- 111
2
votes
0
answers
105
views
Compact Finite Differences for the Heat Equation with Robin Boundary Conditions
I am trying to solve the Heat equation with Robin Boundary condition:
$$ u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$
for $ 0\leq x\leq1$ ...
- 21
1
vote
1
answer
208
views
Nonlinear Robin boundary condition involving square root
If you have a nonlinear second-order boundary value problem where the domain of the problem is $x \in [a,b]$, the boundary conditions imposed are the Robins condition at $x=a$ and the Dirichlet ...
- 163
0
votes
2
answers
268
views
Practical implementation of the discrete compatibility condtion
I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form
$$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
- 83
1
vote
1
answer
301
views
Convergence stall when solving 2D Poisson PDE with pure Neumann boundaries (finite differences)
I recently started coding a small library of 2D PDE solvers (time dependent and time independent), and my first attempt was a 2D Poisson equation of the form:
$$\nabla(\epsilon\nabla\varphi)=\nabla\...
- 83
5
votes
1
answer
322
views
Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?
If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$):
\begin{alignat}{3}
\Delta u(x) &= f(x), &\quad&x\in\Omega \\
\partial_nu(x) &= 0, &\quad&...
- 2,267