All Questions
147 questions
0
votes
0
answers
128
views
Heat Equation for fast source with FiPy
I'm trying to solve the following differential equation with FiPy, basically laser irradiation on a surface
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 11
3
votes
0
answers
53
views
Datasets for inverse heat transfer problems
I was wondering if there is an available, real-life known inverse heat transfer problem dataset to benchmark oneselfs algorithm, as in MNIST for deep learning. Talking about... (well in this case I ...
- 191
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
0
votes
0
answers
62
views
What is the best finite volume method for the following equation?
I'm trying to create a partial differential equation that approximates 1-D climate in a rocky planet's atmosphere, which accounts for energy transport via radiation and convection. I am only ...
- 317
1
vote
0
answers
50
views
How can I apply a mixed boundary condition to a multi-material heat transfer problem using Crank-Nicolson?
I am working on a mixed material model for a melting material and need to enforce both a Dirichlet and Neumann type condition at the interface. Subject to an external surface heat flux at the top of ...
- 11
0
votes
0
answers
38
views
Thermo Hydraulic Mechanical modeling of energy wall slab in Comsol multiphysics
I am currently working on a complex simulation project involving an energy wall slab, and I need assistance in accurately modeling and validating it using COMSOL Multiphysics. Here are the details of ...
0
votes
0
answers
54
views
How to solve the heat equation using the spectral method (Chebyshev's differentiation matrix), with constant flux boundary condition on both sides?
I am trying to solve a 1d heat equation with a constant flux boundary condition on the right-hand side and a zero flux boundary condition on the left-hand side. I've gained a lot of insight from ...
- 1
1
vote
2
answers
121
views
How to handle non bilinear weak form?
I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment.
I modeled the ...
- 63
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:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
3
votes
0
answers
76
views
Confusion between standard finite element and mass-lumping finite element methods
Consider the following equation, subject to homogeneous Neumann boundary condition.
$$
u_t = \Delta u + f(u).
$$
The weak formulation is as follows:
$$
(u_t,w) = (\Delta u, w) + (f(u),w), \quad \...
- 141
0
votes
1
answer
59
views
Time discretisation after splitting a 4th order equation
Suppose we have a fourth-order parabolic PDE
$$\partial_t u + a(t)\Delta( \Delta u) - div( b(x,t)\nabla u) =0$$
We split the equation into two second-order equations by introducing $w = \Delta u$ thus
...
- 3
1
vote
1
answer
338
views
Why does scipy Conjugate Gradient solver fail to converge for non-steady heat equation using Crank-Nicolson method
Could someone please explain why my implementation of the Crank-Nicolson method applied to the non-steady heat equation won't converge? There shouldn't be any nonlinear aspects to my implementation ...
- 13
2
votes
0
answers
114
views
Efficient heat diffusion implementation with varying coefficients
I have the following heat diffusion equation:
\begin{alignat}{3}
\partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\
\partial_n u(t,\...
- 2,892
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
1
vote
1
answer
308
views
2D Heat equation solved with finite element method converges in skewed way
I tried to solve the 2D heat equation with the finite element method, using triangles as elements. Currently generated by a Delaunay triangulation. The base function I'm currently using is basically ...
- 63
0
votes
1
answer
143
views
How to get a normalized gradient with FreeFem++?
I am trying to use FreeFem++ to solve the heat geodesics algorithm.
The algorithm is:
solve $\dot u = \Delta u$ at a specific time $t$.
compute $X = \frac{\nabla u_t}{|\nabla u_t|}$
solve $\Delta\phi ...
- 379
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
0
votes
1
answer
204
views
How do I use pdepe for a first order parabolic PDE with only one boundary condition?
I am trying to use Matlab's pdepe.m to solve the first order parabolic PDE
$$\frac{\partial u}{\partial x}+\frac{\partial u}{\partial x}=x$$
I have not had trouble coding the argument of pdepe @pdefun:...
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
1
vote
0
answers
115
views
Accuracy of the Crank-Nicolson method for non-linear, inhomogeneous heat equation
I am currently coding a solution to the following PDE:
$\frac{\partial T }{\partial t} =\frac{\partial}{\partial \theta}(A(\theta ,\phi )\frac{\partial T }{\partial \theta}) +\frac{\partial }{\partial ...
- 11
2
votes
0
answers
113
views
Conceptual doubt regarding 2D conjugate heat transfer modelling (COMSOL and Mathemtica)
I have been dealing with some conceptual flaws in my understanding of modelling, which I will elaborate herein. I am modelling conjugate heat transfer of a reciprocating fluid, which flows with ...
- 41
0
votes
1
answer
148
views
Problem with my Octave code (unsteady heat equation with FEM)
I want help with my Octave code regarding the unsteady heat equation.
My geometry and mesh are generated with FreeFEM++, so there is no problem with that (I tried it with the steady problem with no ...
- 3
2
votes
1
answer
123
views
Objective function for PDE-constrained boundary control problem in cylindrical coordinates
I'm interested in solving a boundary control problem for an axisymmetric diffusion problem where diffusive fluxes only appear radially. The corresponding problem for a uni-dimensional slab can be ...
- 645
0
votes
1
answer
203
views
Is the Alternating-Directions Implicit method dependent on the space increment?
I am writing an Alternating-Directions Implicit Method for simple 2D diffusion ( \begin{equation*}
\frac{df(x,y,t)}{dt}=D\Delta u
\end{equation*}). Tridiagonal matrices are solved via Thomas ...
2
votes
0
answers
92
views
How to accelerate the computing of implicit finite difference method for heat conduction between two solids
Edit on May 3rd: I have found the problem. Because the difference of between $k_1$ and $k_2$ is huge, a very small time step need to be chosen so that the right green part can "feel" the ...
- 21
2
votes
0
answers
93
views
Numerical scheme for the heat equation on the icosahedral hexagonal grid
I have a predefined grid(like this) that is spawned from a regular icosahedradron. It consists of many hexagons and 12 pentagons (corresponding to icosahedradron vertexes). I can tweak the granularity ...
- 121
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
0
votes
1
answer
94
views
Is this a diffusion equation, or something else？
This is a time fractional PDE，where $0<\alpha<1$.
I would like to know how to describe this equation and where does this equation come from.
- 3
2
votes
0
answers
294
views
Rosenthal equation for multi track
Rosenthal's equation lets one calculate the temperature profile of a moving point heat source analytically for thin and thick plates. For simplicity I use the equation for thick plates defined as:
$$T-...
- 840
1
vote
1
answer
2k
views
How to solve heat equation in spherical coordinates with finite differences?
I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only.
...
- 111
3
votes
1
answer
99
views
"Optimal" domain partitioning in domain decomposition algorithms
When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
- 31
3
votes
1
answer
169
views
Derivation of a parabolic PDE using Alternating Direction Implicit method
I have a very simple question concerning Alternating Direction Implicit (ADI) Scheme.
If I have an equation of the form:
\begin{equation*}
\frac{df(x,y,t)}{dt} = \nabla^2 f(x,y,t) + f(x,y,t)
\end{...
- 319
1
vote
1
answer
236
views
3d schrodinger equation weak form
SCHRODINGER’S EQUATION
$$-ih u_{t}(x,y,z,t) = \frac{h^2}{2m} u_{xx}(x,y,z,t)+ \frac{e^2}{r}u(x,y,z,t)$$
The potential $\frac{e^2}{r}$ is a variable coefficient.
So, let’s take the free Schrodinger ...
user38211
4
votes
1
answer
139
views
Solving geodesics on triangular meshes gives negative distances
I have implemented the heat method for geodesics:
https://www.cs.cmu.edu/~kmcrane/Projects/HeatMethod/paperCACM.pdf
When I run it I am getting a solution that, visually, seems correct:
In this image, ...
- 379
0
votes
0
answers
46
views
The physical meaning of conservative mass in diffusion equation
I am working on 1-D mass transient diffusion in a radial domain (spherical object) using finite volume method. My equation reads
$$\frac{\partial C}{\partial t} = D\left[\frac{\partial^2 C}{\partial r^...
- 1
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
0
votes
1
answer
214
views
Incorporating heat flux into Laplace Equation
I need to find the temperature distribution of a square plate using the Laplace equation by using FDM:
$$ \frac{d^2T}{dx^2} + \frac{d^2T}{dy^2} = 0$$
But there is a heat flux entering from the top ...
- 145
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
0
votes
1
answer
463
views
Verification of order of convergence of Implicit Euler Method to numerically solve Black-Scholes PDE
I'm trying to verify the order of convergence for implicit Euler method to numerically solve Black-Scholes PDE. Theory says that it should be $O(\Delta t + \Delta S^2).$ My code is working absolutely ...
user36184
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(...
0
votes
1
answer
133
views
How to insert a(x) function in non homogeneous parabolic pde for implicit method in Python?
I have the following inhomogeneous parabolic initial/boundary value problem:
$$u_{t}(t,x) = (1-x^{2})u_{xx}(t,x)+u(t,x),$$ for $t \in [0,1]$ and $x \in [-1,1]$
$$u(0,x) = \sin(\pi x),$$ for $x \in [-...
user37062
2
votes
2
answers
219
views
Heat equation in non-dimensional form behaving differently than in usual format
Starting from
$$
c_p \frac{\partial u }{\partial t} = k \nabla^2 u
$$
in a one dimensional domain [0,1] where $c_p$ and $k$ are modeling two different materials:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
- 601
1
vote
1
answer
169
views
Non-Linear advection diffusion with nondifferetiable advection term
I'm looking at Murray's book: Mathematical biology: an introduction , first volume, pag. 404
In particular, I'm interested to solve the following PDE:
$$\partial_t u = \partial_x (\text{sign}(x) u) + \...
- 309
4
votes
2
answers
542
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
1
vote
2
answers
811
views
How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly
My post is structured in four parts:
I give you some information about the context my principal questions refer to.
I will tell you what I believe to know about the Thomas Algorithm. If I am wrong ...
- 11
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
1
vote
0
answers
54
views
Effect of reducing flux consistency order at boundary on convergence order
Consider the 1D nonstationary convection-diffusion PDE
$$
\begin{alignat}{2}
\partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\
f(t) &= \left.\left( a ...
- 451
0
votes
0
answers
89
views
fixed point iteration on DD method
I have to solve the the problem $u_t+\Delta^2u=f(u)$, where $f(u)$ is non-linear, using domain-decomposition method.
My approach is first using fixed point iteration on mixed form i.e to say $u^{k+1}...
- 41
2
votes
0
answers
39
views
Convergent Finite Difference Scheme for Parabolic Equation
Consider the PDE $$u_t = b_{11}u_{xx} + 2b_{12}u_{xy} + b_{22}u_{yy},$$
where $b_{11}, b_{22} > 0$, and $b_{12}^2 < b_1b_2$.
In Strikwerda's book, the ADI scehme \begin{align*}
\left( 1 - \frac{...
- 141
1
vote
0
answers
456
views
Incorporating radiation boundary condition at the edge in finite difference
I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge.
$-k\frac{\partial ...
- 19