All Questions
40
questions
0
votes
0
answers
136
views
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
2
votes
3
answers
178
views
Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?
I'm dealing with the numerical resolution of advection-diffusion-reaction equations.
I've encountered references on CFL conditions for conservation laws as well as advection-diffusion equations.
I'm ...
- 61
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
0
votes
2
answers
134
views
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
1
vote
0
answers
60
views
Stability plot of upward difference implicit time
I am analyzing the stability of 1D convection (advection) equation as shown in the picture. When I derive the equations as shown I want to get rid of the complex number.
I`m asking if those stability ...
1
vote
1
answer
172
views
Finite Difference for Advection Equation With Source
I'm trying to find a convergent finite difference scheme for the PDE
\begin{equation}
\begin{split}
u_t + (x-1) u_x &= (x-1)u, \hspace{.5cm} x \in [0,1] \\
u(x,0) &= 1 \\
u(1,t) &= 1. \\
\...
- 141
2
votes
0
answers
57
views
In sights into why higher order finite differencing method leads faster to instability
I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
- 470
2
votes
0
answers
87
views
How to increase the stability of a DAE solver?
I am trying to solve a set of linear PDEs of the form
$$F\left(\vec{y},\frac{\partial \vec{y}}{\partial x},\frac{\partial^2 \vec{y}}{\partial x^2},\frac{\partial \vec{y}}{\partial t}\right)=0.$$
To ...
- 219
1
vote
0
answers
199
views
Numerically Approximating the Jacobian and Comparing the Eigenvalues With Analytical Form
I am trying to study the stability of numerical discretization schemes using the Jacobian matrix of the residues with respect to the vector of conserved variables.
For a simple diffusion equation ...
- 77
2
votes
2
answers
122
views
Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$
I want to use the Crank-Nicolson scheme to solve the equation
$$u_t = iu_{xx}+2iu$$
Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
- 139
2
votes
1
answer
124
views
Stability Analysis
The partial differential equation,
\begin{align}
\dfrac{\partial f}{\partial x} + a(x)\dfrac{\partial f}{\partial y} = 0
\qquad & f(0,y) = f(L_1,y) = c_0e^{-y} \\
& f(x,0) = c_0 \;,\; f(x,L_2) ...
- 430
2
votes
1
answer
2k
views
Von Neumann's stability analysis on non linear and coupled equations
I'd like to know if is possible to make a Von Neumann's stability analysis on an system of coupled equations, featuring quadratics:
$$\begin{aligned}
\frac{\partial u_1}{\partial t}&=D_1\Delta ...
- 23
2
votes
0
answers
350
views
Von Neumann analysis for coupled PDE's
I am interested in understanding how to perform stability analysis for coupled (to keep things do-able, lets say linear) PDE's
In the case of a single PDE, i understand the logic behind the VN ...
- 121
3
votes
1
answer
210
views
Stability of PDE Discretizations with Multistep Time Discretizations
Let's pretend we have a spatially discretized PDE of the following form:
\begin{align}
\frac{\partial^2 \boldsymbol{u}^k}{\partial t^2} = D\boldsymbol{u}^k
\end{align}
where $D$ can be any form for ...
- 4,278
4
votes
1
answer
1k
views
Von Neumann stability analysis with a constant term
I have a question concerning the von Neumann stability analysis of finite difference approximations of PDEs. There seem to be a wealth of online source explaining the application of this stability ...
- 143