Questions tagged [parabolic-pde]
A partial differential equation that describes diffusion phenomena.
14
questions
14
votes
2
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Periodic boundary condition for the heat equation in ]0,1[
Let us consider a smooth initial condition and the heat equation in one dimension :
$$ \partial_t u = \partial_{xx} u$$
in the open interval $]0,1[$, and let us assume that we want to solve it ...
- 433
5
votes
2
answers
1k
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First order finite volume spatial discretization of the heat equation on an unstructured triangle mesh
Consider a scalar field $u$ on an unstructured triangle mesh which is constant on each face. Let $A_i$ be the area of triangle $T_i$, $N(i)$ the set of triangles sharing an edge with $T_i$, and $L_{...
- 3,889
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votes
2
answers
521
views
Where can I find a good reference for the stability properties of several methods of solving parabolic PDEs?
Right now I have a code that uses the Crank-Nicholson algorithm, but I think that I would like to move to a higher-order algorithm for timestepping. I know that the Crank-Nicholson algorithm is stable ...
- 3,345
0
votes
0
answers
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Analytic vs discrete understanding of PDE
The PDE I am working with:
$$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\
\partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\
\beta(x)>0$$
Integrate the PDE:
$$\int_\Omega \partial_t u=\...
- 216
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votes
2
answers
993
views
Is there a good tutorial or textbook-like source on implementing ENO/WENO with limiters in one (and more than one) dimension?
I've inherited a finite volume code that does a second-order discretization of flux terms for a set of mixed parabolic-elliptic equations with discontinuous diffusion coefficients. The impression I ...
- 30.1k
8
votes
1
answer
339
views
Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde
Suppose I have the parabolic system $$u_t=\nabla\cdot(k(x)\nabla u)+f,\quad (x,t)\in\Omega\times I$$ with Dirichlet boundary conditions $$u=g, \quad x\in\partial\Omega$$ and initial condition
$$u(x,t)...
- 11.9k
6
votes
1
answer
286
views
Do the class of PDEs that lack initial conditions have a name?
I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation.
Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
- 1,000
5
votes
1
answer
3k
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Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method
I want to numerically solve the non-linear diffusion equation:
$$
\frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right)
$$
I want to use ...
- 51
4
votes
1
answer
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Meaning of CFL condition on parabolic problems
I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.
I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
- 1,019
3
votes
2
answers
380
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1D FEM for nonlinear diffusion coefficient
I want to solve with linear finite elements the equation $$\partial_t u = \partial_{x}(a(u)\partial_xu)$$
in the domain $t \in [0,1]$ and $x \in [-L,L]$. Here $a(u)$ is just a function of $u$.
...
- 289
3
votes
1
answer
158
views
Space-time Galerkin of Burgers changes the convection speed
tldr: Can space-time Galerkin schemes applied to convection-diffusion problems lead to effects on the convection velocity?
For time $t\in (0,1)$ and the spatial variable $\xi \in (0,1)$, I am ...
- 3,398
3
votes
3
answers
679
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Algorithm suggestion for PDE - example: heat equation
I want to solve the PDE equation numerically. For this, I started my study with something simple; heat equation
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial^2 x}
$$
with the initial ...
- 319
1
vote
1
answer
152
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Non-Linear advection diffusion with nondifferetiable advection term
I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404
In particular, I'm interested to solve the following PDE:
$$\partial_t u = \partial_x (\text{sign}(x) u) + \...
- 289
1
vote
1
answer
170
views
Error for the finite differences scheme -- Advection equation
Consider the advection equation (1D in space)
$$
\frac{\partial u}{\partial t} + V\, \frac{\partial u}{\partial x}=0
$$
and we solve it numerically on $[0,1]\times [0,1]\ni (t,x)$ using a forward ...
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