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
0
answers
149
views
Population of the coefficient matrix of a linear system Ax=b stemming from the finite differences of an arbitrary geometry
I've been looking into solving a linear system $$Ax=b$$ where $A\in\mathbb{R}$ is the sparse coefficient matrix of size $K\times K$, $b\in\mathbb{R}$ is the right-hand side (i.e., the source term) of ...
- 83
2
votes
1
answer
94
views
Creating nonuniform grids for FDM with multiple points of concentration
If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use:
$$
S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S
$$
where $c=\...
0
votes
0
answers
39
views
Convergence of Modified Crank-Nicolson Scheme
I'm dealing with a particular reaction-diffusion equation having the form
$$ c_t = \alpha \nabla^2 c + F(c,x,t). \tag{1}$$
where $F$ is nonlinear. I would like to solve (1) with a finite-difference ...
- 101
2
votes
0
answers
102
views
Numerical solution for inviscid Burgers' equation seems to have no breaking time?
So I'm trying to use the Lax-Friedrichs method to solve the inviscid burgers' equation with initial condition $$u(x,0) = \sin(x)$$, using
$$u_m^{n+1} = \frac{1}{2}(u_{m+1}^n + u_{m-1}^n) - \frac{\...
- 129
0
votes
1
answer
85
views
My toy Laplace equation solver using finite-difference is unstable and I'm not sure why
I am trying to solve the variable-coefficient Laplace equation $$\partial(\epsilon\partial u) = (\partial\epsilon)(\partial u) + \epsilon\partial^2 u = 0$$using a finite difference scheme:
$$\left(\...
- 417
0
votes
0
answers
109
views
Solving a steady-state PDE
I'm working on trying to solve a steady state PDE in python and I am quite new to python and I am slightly confused on the implementation details. I have an example implementation of solving a non-...
- 43
4
votes
1
answer
148
views
Burger's equation (PDE) does not work with downwind difference?
I'm working on implementing the discretised Burger's equation. I am quite confused as to why it does not work when using a step-function and downwind difference formula. When using a step-function and ...
- 43
1
vote
1
answer
156
views
Finite Difference method, ADI Scheme of Douglas and Rachford
I am trying to implement the ADI scheme of Douglas and Rachford.
For $p(X,Z,t)$, there is:
$$
\begin{gathered}
A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\...
- 11
2
votes
1
answer
340
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
0
votes
0
answers
56
views
ENO-Runge-Kutta discretization
One beginner's question about discretization of a Hamilton-Jacobi equation(non-linear)
$$
u_t = H(u_x)
$$
$u_x$ is discreated with 2nd order ENO-FD
1st order: $D_1^{\pm}u = \pm [u_{x\pm1} - u_x ] / \...
- 1
0
votes
0
answers
45
views
Which numerical method can I use to solve this system of hyperbolic PDEs?
Backround
The mathematical model I am trying to numerically solve models wave propagation inside a cylinder with specific material properties suited for dynamic loading. The cylinder's upper base is ...
3
votes
1
answer
280
views
Finite difference problem
I have a problem to resolve with the Finite Difference method in $[a,b]$:
$$-\frac{d}{dx}(\alpha(x)\frac{du}{dx})= g(x),$$
with $\alpha(x) \in L^{\infty}$ continuous in $]a,c[$ and $]c,b[$ and ...
- 31
0
votes
0
answers
48
views
Is a sort of "z-drift" the result of numerical precision errors in FDM?
Upon solving the 2D wave equation with Neumann boundary conditions $u_x = u_y = 0$ on a rectangular $10 \times 10 \times 10$ grid, I noticed something odd - $u$ seemed to shift upwards with time. This ...
- 133
3
votes
1
answer
186
views
Stability of Euler forward method
I am trying to solve a linear system of ODEs of the form:
$$ \frac{du}{dt} = A u, \quad u(0)=k$$
where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
- 31
1
vote
1
answer
85
views
derivative matrix and the Dirac delta distribution
For a project I'm working on, I was working with the following equation
$$
w(x) = \int k(x,y)v(y)dy
$$
I noticed that if I choose
$$
k(x,y) = -\delta'(x-y)
$$
Then we probably get (I haven't touched ...
- 760