All Questions
Tagged with pde nonlinear-equations
57
questions
1
vote
0
answers
110
views
Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
3
votes
0
answers
173
views
Numerically solving a 6th order non-linear differential equation in Matlab
I've posted yesterday a question about solving a non linear equation : it was not clear so I am reformulating my question.
I am trying to solve a high-order non linear differential equation presented ...
- 33
1
vote
0
answers
69
views
Preserving conservation properties across time-integration schemes
I am interested in deriving an accurate, implicit discretization of a nonlinear advection-diffusion equation
$$
\partial_t f = \frac{1}{v^2} \partial_v J(f,v) \equiv F(f,v) \tag{$\star$}
$$
with flux
...
- 735
0
votes
0
answers
58
views
Discretizing Multi-species Ion Exchange Equations by Finite Volume Method
I'm solving a system of multispecies ion exchange equations (diffusion+drift fluxes) in 1-d spherical domain using finite volume method to obtain the ion concentrations at the next time step. After ...
- 1
3
votes
0
answers
121
views
Strange Picard iteration
I am interested in solving the equation
$$
\begin{aligned} \nabla \cdot\left(\nabla \phi-\frac{\nabla \phi}{|\nabla \phi|}\right) &=0 & & \text { in } \Omega \\ \phi &=0 & & \...
- 601
2
votes
1
answer
229
views
How to discretize a non-linear PDE with boundary conditions and intial value
Consider this non linear PDE:
$$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)
$$
where the 3 functions(...
- 146
1
vote
0
answers
155
views
Solving Laplace equation with constraint on boundary
I have found the following PDE problem in a paper:
Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
- 217
1
vote
2
answers
2k
views
Jacobians with automatic differentiation
I have an objective function F: Nx1 -> Nx1, where N>30000. There are many sparse matrix/tensor multiplications in this function, so taking an analytic Jacobian by paper and pen is cumbersome.
...
- 21
2
votes
1
answer
397
views
Numerical methods that can be written in flux conservative form
I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
- 95
0
votes
1
answer
46
views
Coupled pdes of the first order
May question is about possible approaches to solve the following system
$$
\begin{array}{rcl}
\nabla{n}&=&n\,\mathbf{E},\\
\nabla\cdot\mathbf{E}&=&1-n,
\end{array}
$$
in general with ...
3
votes
1
answer
950
views
Limitations with dynamical systems vs. PDEs?
As a computational scientist, are there limitations to studying dynamical systems — systems of odes in which each state variable evolves with time — compared to studying PDEs?
For instance, it seems ...
- 31
1
vote
1
answer
110
views
Using Kutta Merson on NLS
I'm trying to use the Kutta-Merson to get the same results as in the book Solitons, Nonlinear Evolution Equations and Inverse Scattering - M. J. Ablowitz - pg 140
The author propose using the Kutta-...
- 111
2
votes
1
answer
525
views
How to use Newton-Raphson method to handle nonlinear terms in coupled system of PDEs?
I'm trying to solve the Nonlinear Schrodinger's Equation (NLSE) in 2D using Finite Elements, but I don't know how to handle the nonlinear term. I suppose I have to apply the Newton-Raphson algortihm ...
- 21
2
votes
0
answers
88
views
How to solve $y(x) y'''(x)=f(x)$
I have a PDE of the form $\partial_t y(x,t)+\partial_x(y(x) y'''(x)-f(x))=0$, where $f(x)=\cos(x)$.
Suppose a stable equilibrium exists, and I want to find the steady-state solution $y(x) y'''(x)=f(x)...
- 217
4
votes
1
answer
163
views
References for the nonlinear reaction-diffusion equation using Finite Element Methods
I want to study how to solve the following PDE
\begin{cases}
-\nabla \cdot(\ k(x,y) \ \nabla u \ ) + \beta(x,y)\ u^2 = f(x,y), \ (x,y) \in \Omega \subset \mathbb{R^2} \\
\hspace{0.5cm} u = ...
- 273