All Questions
Tagged with finite-difference nonlinear-equations
44
questions
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
$$u_m^{n+1} = \frac{1}{2}(u_{m+1}^n + u_{m-1}^n) - \frac{\...
- 129
0
votes
1
answer
67
views
Oscillation in non-linear porous flow solved by finite difference
I am trying to solve numerically the flow of a gas through a porous, spherically symmetric body. The non-dimensional equations read:
$$
\frac{\partial\rho}{\partial t}+\tau\frac{1}{r^2}\frac{\partial}{...
1
vote
1
answer
153
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how to solve a coupled nonlinear partial differential equations(Boundary Value Problem)
So far, I have been looking into linear and nonlinear differential equations(Boundary value problem) that look like the following:
$\frac{d^2 \theta}{dz^2} + \sin(\theta) = 0$
I am now familiar with ...
- 35
3
votes
0
answers
207
views
Python code of explicit method of a nonlinear a BVP
I am trying to have a Python code for the following nonlinear BVP:
$$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$ $$N(0,x)=\sin(2\pi x)$$
$$N(t,0)=0 \hspace{3mm}N(...
- 31
1
vote
1
answer
77
views
Once Lyapunov exponents have converged, can they diverge again?
I know the strength of attraction for a single attractor might vary from place to place. Say, if I calculated the Lyapunov exponents for a small portion of the attractor and they have already ...
- 197
2
votes
0
answers
176
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Poisson equation solution in a semiconductor structure
I am trying to solve the $\textbf{1-D}$ Poisson equation for a semiconductor structure at equilibrium (There is no external bias applied).
$\textbf{Background}$
\begin{equation}
\frac{d^2V}{dx^2} = -\...
- 33
0
votes
1
answer
80
views
On solving a first order nonlinear differential equation
It all starts with this Cauchy problem:
$$
\begin{cases}
\sin(2x(t)) -\cos(3x'(t)) = x(t) + x'(t) \\
x(0) = 1 \\
\end{cases}
\quad \quad \text{with} \; t \in [0,10]\,.
$$
Not knowing which way to turn,...
- 113
4
votes
1
answer
207
views
Method to linearize highly nonlinear partial differential equation
I have a set of coupled pdes which I want to solve using finite-difference, of which one is nonlinear. The three linear pdes for quantities $T_f$, $T_s$ and $c$ are convection-diffusion-reaction-like ...
- 81
2
votes
1
answer
229
views
How to discretize a non-linear PDE with boundary conditions and intial value
Consider this non linear PDE:
$$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0 $$
with
$$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x)
$$
where the 3 functions(...
- 146
0
votes
1
answer
216
views
Discretization of a non-linear ODE using FDM isn't grid indepenent
I am trying to solve the ODE :
$\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$
+
using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
2
votes
0
answers
86
views
Linearising Nonlinear Coupled Partial Differential Equations - Alfvénic Diffusion
I am trying to solve the following coupled partial differential equations with a finite difference scheme:
$$\partial_tf+v\partial_zf+\partial_z\frac{1}{W}\partial_zf=0$$
$$\partial_tW+v\partial_zW-\...
- 21
4
votes
1
answer
887
views
Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python
Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788
PDE: u_t = u_xx + u(...
5
votes
0
answers
187
views
2nd-order TVD criteria for flux-limiter
Consider a nonlinear hyperbolic conservation equation:
$$
\partial_{t}u = -\partial_{x}f(u)
$$
The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$
$$
\...
- 95
2
votes
1
answer
397
views
Numerical methods that can be written in flux conservative form
I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
- 95