Linked Questions

22 votes
2 answers
1k views

NDSolve uses different difference order for different spatial derivative when solving PDE

I found something this tutorial for method of line doesn't tell us. Consider the following toy example: ...
xzczd's user avatar
  • 66.2k
19 votes
1 answer
2k views

How to solve the tsunami model and animate the shallow water wave?

Backslide introduced in 9.0, persisting through 13.1. Recently when I was learning differential equations, I noticed there is a shallow water wave equation to model the tsunami propagation. How to ...
LCFactorization's user avatar
16 votes
2 answers
819 views

Conservation of area solving a PDE via finite difference scheme

I have two PDEs that describe the movement of fluid: $h_t + [h^3(1-h)^3((1+\varepsilon h)\sin \theta - \varepsilon h_\theta \cos \theta]_\theta$ = 0 $h_t - [h^3(1-h)^3 \varepsilon h_\theta]_\theta$ = ...
mch56's user avatar
  • 723
9 votes
2 answers
769 views

Numerically solving the KdV equation

Backslide introduced in 9, persisting through 13. I am trying to solve the KdV equation numerically. The following code would work perfectly in version 5: ...
gerald's user avatar
  • 233
7 votes
3 answers
1k views

Time dependent Schrödinger equation in 2D

I have the following Schrödinger equation in $2D$: \begin{cases} \partial_t \Psi(x,t) = V(x,t) \Psi(x,t) \quad x \in [-10,10]^2\\ \Psi(x,0)=\exp( \frac{1}{2} (-x^2+y^2)) \end{cases} where the ...
Vefhug's user avatar
  • 421
7 votes
1 answer
346 views

Instability, Courant Condition and Robustness about solving 2D+1 PDE

After several discussions, I would like to focus on the robustness of solving 2D+1 PDE by considering all suggested methods from @xzczd (see here) I found that the Ratio between the convection term ...
Bob Lin's user avatar
  • 445
6 votes
1 answer
224 views

I understand mol but meet difficulty in understanding how `pdetoode` atumatically generate pde-to-ode-rules by using this strange pattern and rule?

I often solve pdes for my research, and years ago I found pdetoode in this forum is very handy. Although it is a small piece of code, it solves several interesting ...
xinxin guo's user avatar
  • 1,323
5 votes
1 answer
975 views

PDE of real-world system, integral boundary condition

I've stripped all the physical-significance for clarity, but I know that u[x,t] will be everywhere positive and continuous. here are the equations in Mathematica code: ...
Paul_A's user avatar
  • 487
5 votes
3 answers
317 views

Problems with solving PDEs

I am using NDSolve to solve the two equations: ...
user55777's user avatar
  • 671
4 votes
2 answers
431 views

Using NDSolveValue for Solving a parabolic PDE numerically

Inspired by this question I am trying to solve the following PDE numerically on $x \in [-3, 3]$ and $t \in [0, 0.5]$ using NDSolveValue: $$ \frac{\partial p}{\partial t} = (12x^2-4) p + \left[4x(x^2-1)...
Eldar Sultanow's user avatar
4 votes
1 answer
197 views

Steady state solution (1D) of nonlinear dispersal equation

Now I'm interested in the equation $$\frac{\partial }{\partial x}\Bigl(\text{sgn}(x) u \Big) +\frac{\partial}{\partial x} \Bigl[ u^2 \frac{\partial u}{\partial x} \Bigr] =0$$ with boundary conditions $...
Vefhug's user avatar
  • 421
4 votes
1 answer
224 views

Different results from NDSolve of v9 and v11

When using NDSolve to solve 2 pdes with different version of Mathematica, I obtained totally different results. The code is as follows. ...
Nobody's user avatar
  • 823
3 votes
1 answer
141 views

Inactive[Grad] is lost when NDSolve parses PDE

I encountered this when trying to solve the PDE mentioned here. I've transformed the equation to the following: ...
xzczd's user avatar
  • 66.2k