Questions tagged [finite-difference]
Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.
892
questions
3
votes
1
answer
270
views
Why does the FDM give a correct solution to a PDE with a discontinuous initial condition?
I was solving the dimensionless wave equation:
$$ u_{xx} = u_{tt} \tag 1$$
with the initial conditions:
$$ u(x,0) = 0 \tag 2 $$
$$ u_t(x=0,0) = v_0 \tag 2 $$
$$ u_t(x>0,0) = 0 \tag 3 $$
and the ...
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
2
votes
0
answers
112
views
Finite difference scheme to 1D wave equation with Dirac Delta forcing term
I am trying to simulate the following 1-dimensional wave equation with trivial initial conditions and a inhomogeneous Dirac delta function:
$u_{tt} - c^2 u_{xx} = \delta(x - x')\delta(t - t'), \ u(0, ...
- 121
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
1
vote
0
answers
107
views
How to implement boundary conditions for the Thomas algorithm
For my variable $U(t,x)$, I have implemented the thomas algorithm with $U_j^i$:
$$ a(x)U_{j-1}^{i+1}+ b(x)U_j^{i+1} + c(x)U_{j+1}^{i+1} = d(x)U_j^{i} $$
Then $\textbf{A}$ is a tridiagonal vector with ...
0
votes
0
answers
53
views
Prof A. Stanoyevitch's finite difference based PDE matlab code
Where can one find Prof A. Stanoyevitch's finite difference based PDE matlab code? They have a book on such a topic but can't find the accompanying code.
Is it well received? It's not commonly talked ...
- 317
2
votes
0
answers
50
views
Upwind scheme flux conservation not satisfied in 2D
I have read that the upwind scheme is flux conservative. Then by my understanding, in the absence of sinks/sources and with absorbing boundaries, the amount of the quantity leaving through the ...
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 ...
2
votes
1
answer
132
views
Can the Runge-Kuta algorithm help in reducing numerical dispersion and anisotropy when using the FDM to solve the 2D wave equation? [closed]
I am currently studying the effects of group velocity on the finite difference solution of the wave equation. Most of what I learned is from this source. I understand that high frequency components in ...
- 45
1
vote
1
answer
95
views
Reference request: graph Laplacian approximation for domains/manifolds
Is there any reference on solving the heat equation on irregular geometries via creating a mesh and using the graph Laplacian instead of FEM techniques? (Convergence to real solution).
That is to say, ...
- 181
1
vote
0
answers
102
views
Can I reduce my simulation error with a staggered grid, postprocessing and compatibility equation feedback?
What I did
Using the finite difference method, I solved with a certain amount of error the following system of hyperbolic partial differential equations in cylindrical coordinates (the problem is ...
0
votes
0
answers
47
views
2nd-order backward difference approximation for temporal terms of Euler–Bernoulli beam
I am trying to solve the Euler–Bernoulli beam numerically in structural dynamics analysis:
$$\frac{\partial^2w(x,t)}{\partial t^2}= \frac {q(x,t)}{\mu(x)}-\frac {1}{\mu(x)}\frac {\partial^2}{\partial ...
1
vote
1
answer
139
views
Best finite difference scheme in 2D for the mixed derivative
The are good methods to deduce finite difference schemes for derivatives of functions of one variable. But how to get a good one for the mixed derivative of a function of two variables $u=u(x,y)$, ...
- 113
1
vote
0
answers
125
views
How to vectorise numerical differentiation
I have a 2-D matrix with 2 spatial coordinates and I want to be able to vectorise the process of numerically differentiating with respect to its 2 coordinates, rather than just looping along the rows ...
0
votes
2
answers
2k
views
When can I use finite differences for differentiation?
Finite differences are usually used to integrate ODE's and PDE's. However, sometimes they can be used for differentiation which I illustrated simply by using the Matlab code below to differentiate the ...