All Questions
Tagged with finite-difference stability
58
questions
0
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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:
$$\frac{\partial^2w(x,t)}{\partial t^2}= \frac {q(x,t)}{\mu(x)}-\frac {1}{\mu(x)}\frac {\partial^2}{\partial ...
0
votes
2
answers
2k
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When can I use finite differences for differentiation?
Finite differences are usually used to integrate ODE's and PDE's. However, sometimes they can be used for differentiation which I illustrated simply by using the Matlab code below to differentiate the ...
0
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0
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45
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Which numerical method can I use to solve this system of hyperbolic PDEs?
Backround
The mathematical model I am trying to numerically solve models wave propagation inside a cylinder with specific material properties suited for dynamic loading. The cylinder's upper base is ...
3
votes
1
answer
186
views
Stability of Euler forward method
I am trying to solve a linear system of ODEs of the form:
$$ \frac{du}{dt} = A u, \quad u(0)=k$$
where $A$ is a 2x2 matrix and $u(t)$ is a 2x1 column vector. I want to solve this numerically, using ...
- 31
2
votes
1
answer
146
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Can this finite difference dispersion be eliminated somehow?
I am trying to solve the wave equation
$$ {\partial ^2u(t,x) \over \partial x^2} = {\partial ^2u(t,x) \over \partial t^2} \tag1 $$
with the following boundary and initial conditions:
$$ {\partial u \...
1
vote
0
answers
144
views
Stability for the 2d diffusion equation
I'm reading this set of notes on numerical stability of the diffusion equation. I can't seem to derive equation (7.17) from the previous equation. I've tried using $\sin^2(\theta)=\frac{1-\cos(2\theta)...
- 760
2
votes
0
answers
69
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Literature request for pinning the corner singularities in finite differences
Corner singularities cause instability of finite difference solutions. They occur at the intersection of boundary conditions. I was recently going through an online video course that was explaining ...
0
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136
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2D wave equation is numerically unstable using Finite Difference Method
I'm working with simulating both the heat and wave equation in 2D in a Python code. When simulating the heat equation, I learned about the CFL which I used to get a numerical stable solution.
I found ...
- 101
0
votes
2
answers
268
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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
4
votes
1
answer
220
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Stepping over a rapid oscillation in advection
As a model of a more complex problem, I am studying a linear advection equation with a rapidly oscillating velocity
$$
\partial_t u + a \cos(\omega t) \partial_x u = 0
$$
with initial condition $u(x,0)...
- 735
4
votes
1
answer
566
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Computing second derivatives with Neumann boundary condition
I am implementing a finite difference method for a PDE with a Neumann boundary condition. I will simplify my question to a single dimension.
Suppose I have a PDE
$$\frac{\partial u}{\partial t} = \...
- 41
2
votes
0
answers
133
views
About the the stability of using an explicit scheme on the heat equation
Before I get to the heat equation I'd like to talk about the advection equation.
Descritize that with FD in time and BD in space:
\begin{equation}
\dfrac{u^{n+1}_i - u^{n}_i}{\Delta t} + v \dfrac{u^{n}...
- 506
4
votes
1
answer
150
views
Stability analysis simplification for PDE
I have the nonlinear PDE
$$\frac{\partial U(z,t)}{\partial t} + A(U)\frac{\partial U(z,t)}{\partial z} + B(U)U(z,t) + C(z,t) = 0,$$
where $A(U)$ and $B(U)$ are guaranteed to be real and positive.
I ...
- 43
1
vote
2
answers
254
views
Stability condition FCTS method
The FTCS method comes from the discretization of a diffusion PDE like this:
$$
a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t}
$$
If I have the ...
- 23
0
votes
2
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134
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Stability of the Forward-Time Central Space method, section 9.3 in LeVeque
I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
- 146