Questions tagged [pde]
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.
916 questions
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adjoint method + complementarity problems
Background.
The adjoint methods give a way to evaluate the derivative of some quantity of interest $J$ depending on a field $u$ with respect to a set of parameters $p$ when $u$ and $p$ are related by ...
- 11.1k
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How to find Eigenfunctions and values for PDE (Helmholtzequation)?
Task: Given some spacial domain in 2D (e.g. a triangle), Dirichlet boundary conditions find the Eigensolutions/Eigenvectors $k$ of the Helmholtz-equation.
$$
\Delta \phi(x,y)+k^2\phi(x,y)=0
$$
Problem:...
0
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0
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Methods to Solve PDE of the Form $u_t = D u_{xx} - c(t) u_x + F(u)$
I am trying to solve the following PDE,
\begin{align}
\begin{split}
u_t &= D u_{xx} - c(t) u_{x} + F(u) \\
u(t, 0) &= u(t, L) \\
u(0, x) &= u(T, x) \\
c(...
- 163
1
vote
1
answer
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Why do solutions of 1-D Advection Equation using Crank-Nicolson Scheme negative?
I'm trying to approximate the solutions of this 1-D Linear Advection Equation using Crank-Nicolson Scheme, and after some loop of repeating it in Python, the values return to negative. I didn't come ...
- 11
0
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2
answers
130
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Backward Euler Explanation for a beginner
Can someone explain Backward Euler in the context of a transient thermal problem in 2-dimensions for simplicity, lets say I have a weak form of the thermal problem , i would like to understand how to ...
- 1
0
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0
answers
56
views
traveling wave diverged in 3d using the same 1d time stepping
A dissipation free traveling wave u(x-ct) with periodic boundary conditions is properly computed in 1d regular Galerkin FEA. With the same time stepping (runga kutta, BDF etc), its 3d regular Galerkin ...
- 325
2
votes
0
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Differentiable Level Set Method for Interface Growth with Overgrowth over Inactive Regions
I'm trying to simulate an interface evolution problem using a level set method, with the key constraint that the entire simulation must be differentiable (end-to-end for later use with optimization/...
- 21
1
vote
1
answer
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Proper boundary conditions and numerics of a traveling wave solution
To get a traveling wave solution of $u_t+cu_x=0$ on $0<x<1: u=u(x-ct)$, periodic boundary conditions are applied at $x=0$ and $x=1$ i.e., $u(0)=u(1)$, $u_x(0)=u_x(1)$. Is that sufficient to get ...
- 325
0
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MATLAB numerically solve coupled nonlinear PDEs of laser
I try to solve two coupled nonlinear pdes, but it fails to converge. I'm new to PDEs, and have trouble to reproduce the results of the paper. The equations are in the picture below:
They are all ...
- 1
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61
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Numerical methods for non-incoming boundary condition for PDEs
I am discretizing a PDE with a non-incoming boundary condition, defined as
$$
n(x)\cdot v <0
$$
I wonder is there any reference on now to numerically implement this in a simple way? The current ...
- 101
1
vote
0
answers
52
views
Accounting for boundary conditions while using fixed-point iteration to solve a nonlinear heat equation
I'm trying to solve a nonlinear heat equation
$$
\frac{\partial T}{\partial t} = f(T,z,t)
$$
where $$
f(T,z,t) = \alpha(T)\frac{\partial^2 T}{\partial z^2}+\beta(T)\left(\frac{\partial T}{\partial z }\...
- 11
1
vote
0
answers
58
views
Centered Finite Difference Scheme for Stochastic Wave Equation
I am attempting to simulate a 2D damped wave equation with a stochastic forcing term:
\begin{align}
&\ddot{u}+\alpha \dot{u} + \beta u-c^2 \nabla^2 u=\dot{f},\\
&u(\mathbf{r}, 0) = \dot{u}(\...
- 131
2
votes
1
answer
66
views
Numerical treatment of the drift term in Gross-Pitaevskii equation
I am numerically trying to solve the following Gross-Pitaevskii equation that describes the dynamics of an impurity coupled with a one-dimensional weakly interacting bath:
\begin{align}
i \frac{\...
- 41
3
votes
1
answer
161
views
How to carry out Von Neumann stability analysis for a coupled system?
Background:
There is already a question here:
CFL criterion for a PDE
Although the accepted answer makes intuitive sense, it does not seem mathematically rigorous.
I realize that this original ...
- 183
0
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1
answer
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CFL criterion for a PDE
Given the PDE
$\frac{\partial}{\partial t} \frac{\partial^2 \Psi}{\partial x^2}+ \beta \frac{\partial \Psi}{\partial x} + U \frac{\partial}{\partial x} \frac{\partial^2 \Psi}{\partial x^2}=0$
with ...
- 183