All Questions
147 questions
14
votes
2
answers
19k
views
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 ...
- 443
11
votes
3
answers
377
views
What is the current state of the art in solving higher dimensional parabolic PDEs (multi-electron Schrödinger equation)
What is the current state of the art for solving higher dimensional (3-10) parabolic PDEs in the complex domain with simple poles (of the form $ \frac{1}{|\vec{r}_1 - \vec{r}_2|}$) and absorbing ...
- 1,155
10
votes
2
answers
528
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 ...
- 3,355
10
votes
2
answers
1k
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 ...
- 30.4k
9
votes
1
answer
2k
views
Example of a PDE model with nonlinear Dirichlet boundary conditions
Is there any application for PDEs with nonlinear Dirichlet boundary conditions? That is, I am looking for an example of a partial differential equation for a state $u$ posed on a domain $\Omega$ with $...
- 3,418
9
votes
1
answer
1k
views
Optimal use of Strang splitting (for reaction diffusion equation)
I made a strange observation while computing the solution to a simple 1D reaction diffusion equation:
$\frac{\partial}{\partial t}a=\frac{\partial^2}{\partial x^2}a-ab$
$\frac{\partial}{\...
- 2,161
8
votes
1
answer
192
views
How to recreate this result (from a book)?
The result I'm interested in is found within "Synchronization: A Universal Concept in Nonlinear Sciences" page $333$ figure $14.3$. The peculiar fragment is also provided at the end of this post.
So ...
- 191
8
votes
1
answer
374
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
$$u(x,t)...
- 12.2k
8
votes
1
answer
550
views
Computing geodesic distances with diffusion
I am trying to solve an APSP (All-Pair Shortest Path) problem on a weighted graph.
This graph is actually a 1, 2 or 3 dimensional grid, and the weights on each edge represent the distance between its ...
- 131
7
votes
3
answers
981
views
analyze stability on a nonuniform grid
Assume you have a stability constraint between the space distance in time and space, for example, with an explicit Euler method for $u_t=u_{xx}$ we know $\tau\leq h^2/2$. That is, one can do stability ...
- 1,216
6
votes
1
answer
302
views
Do the class of PDEs that lack initial conditions have a name?
I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation.
Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
- 1,000
6
votes
1
answer
705
views
eigenvalue analysis vs fourier analysis for stability and their equivalence
I have a question regarding stability analysis for constant coefficient pde. Suppose I am looking at the pde $$u_t= au_{xx}$$ So the first approach is to compute the amplification factor, and this is ...
- 1,216
5
votes
1
answer
4k
views
Meaning of CFL condition on parabolic problems
I've been studying this FEM theory and for the parabolic problems, there's the analysis of stability of the $\theta$-method.
I followed the analysis and they get this CFL (Courant-Friedrich-Lewy) ...
- 1,039
5
votes
2
answers
4k
views
Crank-Nicolson method for solving nonlinear parabolic PDEs
Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ?
I tried to apply this method for solving such system but ...
- 281
5
votes
2
answers
366
views
Continuous vs discontinuous space-time FEM
What are some reasons for approximating a (e.g. parabolic) PDE using the space-time method, with continuous finite elements in time, vs discontinuous finite elements in time?
Are there e.g. ...
- 259
5
votes
2
answers
1k
views
First order finite volume spatial discretization of the heat equation on an unstructured triangle mesh
Consider a scalar field $u$ on an unstructured triangle mesh which is constant on each face. Let $A_i$ be the area of triangle $T_i$, $N(i)$ the set of triangles sharing an edge with $T_i$, and $L_{...
- 3,969
5
votes
1
answer
3k
views
Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method
I want to numerically solve the non-linear diffusion equation:
$$
\frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right)
$$
I want to use ...
- 51
5
votes
1
answer
2k
views
The effect of the boundary condition on the convergence of the fdm scheme
I know the boundary condition is usually a tricky question. However, I am testing a finite-difference scheme for the equation of the form $$u_t=a(x)u_{xx}$$ that I know the analytical solution of. So ...
- 1,216
5
votes
1
answer
5k
views
How to add reaction and source terms to a diffusion PDE solver written with MATLAB's pdepe function?
I have the following system of equations which I'm trying to solve using Matlab's pdepe solver.
The 1-D spherical heat diffusion equation with heat generation (...
- 325
5
votes
1
answer
184
views
Stabilization of solution to one-dimensional system of PDE
I am trying to solve numerically next PDE system:
$$\frac{\partial c}{\partial t}=\epsilon\frac{\partial}{\partial x}(\frac{\partial c}{\partial x}+\rho\frac{\partial \varphi}{\partial x}+\frac{vc}{1-...
- 51
5
votes
1
answer
90
views
regularity of a solution and its affect on the global error
I am solving the following equation $u_t=x^2u_{xx}+\frac{x-y}{T-t}u_y$ with an initial data $u_0=max(y-C,0)$ for some $C$ in the domain of the numerical method. In time I am solving that on $[0,T]$. ...
- 1,216
5
votes
1
answer
107
views
Prediction of sphere (i.e. roast) core temperature heated in an oven
The real-life problem
Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
- 161
5
votes
1
answer
560
views
Does ADI/Split-operator change the stability properties of the Crank-Nicholson method?
I'm using the Crank-Nicholson method to solve the time-dependent Schrödinger equation with the split-operator method. I'm getting some weird results that are probably the result of a bug somewhere in ...
- 3,355
4
votes
2
answers
343
views
Is there a simple way to add a sparse matrix to an LU decomposition of a dense matrix?
I am solving a parabolic equation in the form:
$$
\left( {M \over\tau_j} + A \right) u^{j+1} = M f^j + {u^j \over \tau_j},
$$
where $A$ and $f$ are a dense stiffness matrix and the right hand side of ...
- 183
4
votes
1
answer
1k
views
Is the heat equation with Neumann boundary conditions well-posed?
For example I consider a heat equation that I want to solve numerically : $$u_t=u_{xx},$$ In order to have a uniqueness on a computational bounded domain I have to have boundary condition specified ...
- 1,216
4
votes
1
answer
139
views
Solving geodesics on triangular meshes gives negative distances
I have implemented the heat method for geodesics:
https://www.cs.cmu.edu/~kmcrane/Projects/HeatMethod/paperCACM.pdf
When I run it I am getting a solution that, visually, seems correct:
In this image, ...
- 379
4
votes
2
answers
543
views
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 $u$.
...
- 309
4
votes
1
answer
230
views
Boundary value technique for heat equation
My heat equation is
$$
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad x \in [0,1], \quad t \in (0,0.1]
$$
with initial condition $u(x,0)=\sin(\pi x)$ and homogeneous ...
- 263
4
votes
1
answer
923
views
Backwards Difference Implicit Method for Nonlinear Parabolic PDE in Python
Original Stack Overflow Question: https://stackoverflow.com/questions/65683788/indexerror-index-31-is-out-of-bounds-for-axis-1-with-size-31?noredirect=1#comment116218335_65683788
PDE: u_t = u_xx + u(...
4
votes
2
answers
440
views
choice of the norm for Crank Nicolson stability estimate
I have a variable coefficient pde of the form $$u_t=c(t,x)u_{xx}, t\in [0,T], x\in [0,1]$$ with initial data $u_0=u(0,x)$ and $c(x,t)\in C([0,T]\times[0,1])$. I use three point discretization for the ...
- 1,216
3
votes
3
answers
712
views
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
3
votes
1
answer
193
views
Type of Rosenbrock method by its coefficients
A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients:
...
- 226
3
votes
1
answer
99
views
"Optimal" domain partitioning in domain decomposition algorithms
When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
- 31
3
votes
1
answer
272
views
intuition behind the different discrete norms for Crank Nicolson
I am solving a heat equation $u_t=Au$ with Crank-Nicolson finite-difference method and $A$ is a usual discretization matrix for $u_{xx}$ term. I want to tell something about the whole error vector ...
- 1,216
3
votes
1
answer
79
views
semiboundedness of the operator and it is affect on stability
I remember seeing in the book by Kreiss "Time-dependent partial differential equations and their numerical solution" that if some elliptic differential operator satisfies $$(Lu,u)\leq K(u.u)$$ for the ...
- 1,216
3
votes
1
answer
114
views
Nondimensionalization of a multi-component chemical diffusion equation
Edit I've modified the equations because they were wrong and I added the whole system, as asked by @Wolfgang
I am trying to nondimensionalize a system of partial differential equations similar to 2nd ...
- 319
3
votes
1
answer
170
views
Derivation of a parabolic PDE using Alternating Direction Implicit method
I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme.
If I have an equation of the form:
\begin{equation*}
\frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t)
\end{...
- 319
3
votes
1
answer
363
views
steady state solution from parabolic problem vs solution of elliptic problem
My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde
Suppose I solve the parabolic PDE:
$u_t = \Delta u + f(x,t)...
- 445
3
votes
1
answer
173
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 ...
- 3,418
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
3
votes
0
answers
76
views
Confusion between standard finite element and mass-lumping finite element methods
Consider the following equation, subject to homogeneous Neumann boundary condition.
$$
u_t = \Delta u + f(u).
$$
The weak formulation is as follows:
$$
(u_t,w) = (\Delta u, w) + (f(u),w), \quad \...
- 141
3
votes
0
answers
160
views
Form of nonlinear diffusion equation
Consider the following nonlinear diffusion problem,
$$
\frac{\partial u}{\partial t} = x^{-2}\frac{\partial}{\partial x}\left(x^2 u^4 \frac{\partial u}{\partial x}\right), \quad 0 < x < 1
$$
We ...
- 645
3
votes
0
answers
108
views
Correct approach for thermal finite element simulation of layered assembly
I would like to optimise the heat transfer on a PCB. Several dies are on the top and cooling air is going through the fins in heat sink on the bottom. The assembly consists of several layers like ...
- 570
2
votes
3
answers
553
views
Flux sign and face normal confusion in finite volume method
I implemented a solver for the 2D steady-state heat equation (without heat generation and homogeneous material) $\nabla. (k\nabla T) = 0$, using finite volume method, however, I am having some ...
- 304
2
votes
3
answers
2k
views
Applying the method of lines to parabolic PDEs: references and software
Could you please advise some literature about the numerical method of lines (MOL) for parabolic PDEs? It is a method of solving PDEs with discretizing only by space but not by time. A system of ODEs ...
- 281
2
votes
2
answers
468
views
Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?
I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions.
So far, I have found the problem solved analytically in one ...
- 61
2
votes
2
answers
438
views
FEM for a nonlinear parabolic PDE
I'm looking to numerically compute the solution to
$$ k(x,u) \partial_t u - \Delta u = f \quad\quad\text{ in } \Omega \times [0,T]$$
where $k$ is a continuous but nonlinear (in $u$) real-valued ...
- 21
2
votes
2
answers
138
views
how to approach time zero when the equation is not defined at that point
Some background: define a process $Y_t=\frac{1}{t}\int_0^tX_sds$, where $X_t$ is a standard Brownian motion. Then I define a function $u(t,X_t,Y_t)$ and require it to be a martingale. Thus, by ...
- 1,216
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
2
votes
1
answer
311
views
Lumped matrices in thermal analysis using finite elements
The governing equation of the transient heat transfer problem is
$$C \frac{dT}{dt}+K T = Q$$
$C$ is the heat capacity matrix. $K$ is the thermal conductivity matrix. $T$ is the temperature vector. $...
- 840