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Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.

7
votes
It's a bit easier to see if you write your equation in the a semi-discretised system of the form $u^{\prime}(t) = F(u(t))$ and with the application of the $\theta$-method and approximating $u^{\prime} …
answered Sep 11 '15 by boyfarrell
26
votes
2answers
I am not very familiar with the common discretization schemes for PDEs. I know that Crank-Nicolson is popular scheme for discretizing the diffusion equation. Is also a good choice for the advection te …
asked Feb 28 '13 by boyfarrell
5
votes
2answers
This might be a naive question, but when applying a implicit discretization to a PDE with a source term, should the source be averaged in time? For example if we take the diffusion equation with a …
asked May 29 '13 by boyfarrell
19
votes
Starting with the advection equation is conservative form, $$ \frac{\partial u}{\partial t} = -\frac{\partial (\boldsymbol{v} u)}{\partial x} + s(x,t) $$ The Crank-Nicolson method consists of a time …
answered May 28 '13 by boyfarrell
2
votes
1answer
I wish to solve a coupled system of non-linear equation of this form, $$ u_t = -(\mathcal{F})_x + f(x,u,w) \\ w_t = \mathcal{F} + g(x,u,w) $$ by stepping the equations forward in time. The first eq …
asked Jul 10 '13 by boyfarrell
0
votes
(I realised the issue, after having a discussion offline). The answer is yes, and the equations are correct. The error is in assuming that the electron and hole densities should remain constant when …
answered Mar 18 '13 by boyfarrell
5
votes
2answers
I am trying to understand an extra terms that appears when I derive the drift-diffusion equations for semiconductors. The extra term (see below) comes from applying the chain rule to the advection ter …
asked Mar 18 '13 by boyfarrell
14
votes
3answers
In semiconductor simulation, it is common that the equations are scaled so they have normalised values. For example, in extreme cases electron density in semiconductors can vary over 18 order of magni …
asked May 25 '13 by boyfarrell
2
votes
Predator-Prey could have the properties you want. Wikipedia has a good description, https://en.m.wikipedia.org/wiki/Lotka–Volterra_equations Alternatively you could solve very simple decay ODE with …
answered Oct 29 '15 by boyfarrell
2
votes
For the advection-diffusion equation you can't just apply a Neumann boundary condition (the 'gradient zero condition' as you called it) because there is are two components to the flux: advection and …
answered Jan 19 '14 by boyfarrell
32
votes
2answers
I am trying to solving the advection equation but have a strange oscillation appearing in the solution when the wave reflects from the boundaries. If anybody has seen this artefact before I would be i …
asked Mar 4 '13 by boyfarrell
2
votes
1answer
Say we have a discretised a coupled nonlinear system of two PDEs to give a system of ODEs which approximates the original system, $$ \frac{\partial u}{\partial t} = F_1(t,\boldsymbol{u,v}) \\ \frac{ …
asked Sep 25 '13 by boyfarrell
24
votes
1answer
I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density …
asked Mar 5 '13 by boyfarrell
6
votes
3answers
Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term), $$ \frac{\partial \phi}{\partial t} = \frac{\partial}{ …
asked Mar 8 '13 by boyfarrell