All Questions
31
questions
0
votes
1
answer
68
views
Local truncation error of given implicit 1-step scheme
I'm given the 1-step implicit scheme $$y_{n+1} = y_n + \frac{h}{6}[4f(t_n, y_n) + 2f(t_{n+1}, y_{n+1}) + hf'(t_n, y_n)],$$
where $y'(t) = f(t, y)$, and I'm seeking the scheme's local truncation error. ...
- 1
2
votes
1
answer
81
views
Differential Equation with Forced Behavior
I'm attempting to solve a strange differential equation problem. My goal is to know if there are kinds of ODE solver packages to solve this kind of problem.
I'm solving a 1D Partial Differential ...
- 317
2
votes
1
answer
367
views
Solving stiff ODEs: Dealing with Jacobian terms which take too long to compute with finite differences
I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
- 317
0
votes
0
answers
62
views
What is the most common loss function used with collocation methods for differential equations
I was looking at the Cheney and Kincaid book (6th edition) on numerical methods, with respect to collocation method for differential equations. Now for linear systems of ODES, collocation is just a ...
- 297
3
votes
0
answers
102
views
Solving PDEs: What is the best way to deal with non-banded/dense jacobians?
I have a system of PDEs describing atmospheric chemistry and transport. I use finite-differences to make my system of PDEs into a system of ~10,000 ODEs. I then integrate the ODEs forward in time with ...
- 317
0
votes
0
answers
109
views
Nondifferentiable coordinate transforms
Suppose that we have coordinates $u=u(x,y)$ and $v=v(x,y)$ in $\mathbb{R}^2$ so that $v$ is not differentiable when $u(x,y)=u_0$ where $u_0$ is a constant. Can we solve a differential equation, such ...
- 331
2
votes
2
answers
775
views
Solving a 1D diffusion equation with linear and nonlinear source terms
I would like to numerically solve the following equation: $$\frac{\partial \rho (z,t)}{\partial t} = B(N_D \rho (z,t) + \rho(z,t)^2) + D \frac{\partial^2 \rho (z,t)}{\partial z^2}$$
with the boundary ...
- 29
0
votes
1
answer
135
views
Why is modeling a physical system with ODEs sufficient?
I've read a few papers in dynamical systems where the model equations are sets of ODEs, with the state space, say, the spatial variables x, y, z, and an angle variable phi all evolving forward in time....
- 13
3
votes
1
answer
252
views
Good reference on the implementation and limitations of SDIRK methods
For the solution of many PDE, implicit high-order time integration schemes are required. I am specifically interested in schemes that do not require a constant time step.
I am well acquainted with ...
- 1,157
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
2
votes
1
answer
60
views
Discretization with non-constant matrix containg entries form unknown vector
Consider a system of PDEs
$$
\begin{cases}
u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\
c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u
\end{cases}
$$
with some boundary conditions. Here, $D(...
- 226
3
votes
0
answers
110
views
How to account for the interface between two different phases in a discretized diffusion model?
I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
- 91
6
votes
2
answers
309
views
Why naively chopped finite difference matrix works for different ODE boundary conditions
We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
- 303
1
vote
2
answers
196
views
Learning differential equations: a textbook
I am a Computational Biology master's student and I would like to study differential equations (both ODEs and PDEs).
I tried to have a look at the available textbooks but what I found is either too ...
- 404
1
vote
1
answer
75
views
Examples of finding eigenfunctions of coupled DEs
I am looking for examples of numerically finding eigenvalues/eigenfunctions of coupled DEs. If anyone is able to point me towards any examples, preferably with code included, it would be much ...
- 147