All Questions
137 questions
1
vote
1
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143
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Derivative Approximation from Trigometric Interpolation vs Polynomial Interpolation
In the classical finite difference method one takes the derivatives of interpolating polynomials and derives the finite difference coefficients from those. The polynomial degree and consistency can be ...
- 2,892
2
votes
1
answer
125
views
Solving advection-diffusion equation on non-rectangular domain
I am trying to solve a PDE similar to the advection-diffusion equation:
$$
\frac{\partial T}{\partial t} + (\vec{u} \cdot \nabla)T = D \Delta T
$$
(where $\vec{u}$ is a known advecting vector field) ...
- 121
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
1
vote
0
answers
85
views
Eigenvalue Problem with Pseudospectral Chebyshev Polynomials
I am solving a linear 4th Order Eigenvalue ODE (Euler-Bernoulli Beam):
$$
{\frac{d^{4}w}{dx^{4}}} = - \alpha {\frac{d^{2}w}{dx^{2}}}
$$
The method I used was to apply a truncated spectral expansion ($...
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
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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
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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
6
votes
2
answers
742
views
What is the advantage of using a particular RK Scheme?
The Wikipedia article on Runge-Kutta Methods lists several examples of each order. My question is, are there any particular advantages using one particular scheme over another of the same order?
I ...
- 193
0
votes
0
answers
81
views
How to correctly implement boundary conditions with Chebyshev differentiation matrix?
In my code I am able to implement zero Dirichlet boundary conditions, however, in my opinion my method does not seem to be as smooth and effective as possible. I am solving a Poisson's equation with ...
- 121
1
vote
0
answers
71
views
Where is the issue in my nonlinear PDE solver using pseudo-spectral methods with Chebyshev collocation?
This is a repost from another community, I think the question would be better here.
This bug has been haunting me for a while and would love new pair of eyes to help, not sure if this is the right ...
- 121
2
votes
0
answers
82
views
Preventing an Overflow in Exponential Integrating Factor
The following is the Discrete Spectral Vorticity Evolution PDE for incompressible flow:
$$ \frac{\partial \Omega_{pq}}{\partial t} = \nu \left( \frac{\partial^2 \Omega_{pq}}{\partial x^2} + \frac{\...
- 193
2
votes
1
answer
409
views
What is the correct way to implement Neumann type boundary conditions for solving a PDE using Chebyshev's collocation method?
I am studying spectral methods for solving PDEs numerically. I finished a chapter that explains how to use Chebyshev's collocation method to solve them. Though the explanation in the book is quite ...
- 45
2
votes
1
answer
411
views
Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ 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
1
answer
191
views
How can we calculate mixed derivatives numerically using the Chebyshev derivative matrix?
Using the Chebyshev derivative matrix $D$, we can numerically approximate the first and second derivative of a function by doing matrix multiplication:
$${df(x) \over dx} = Df(x) \tag 1$$
$${d^2f(x) \...
1
vote
1
answer
150
views
in Finite Element, which approximation requires less number of unknowns: B-splines vs Shape functions vs Spectral Elements
In Finite Element Analysis, one requires to approximate the solution in terms of basis functions. My questions is, which method in general is better? by 'better', I mean which involves less number of ...
5
votes
1
answer
903
views
Taking derivative using FFT
I would like to calculate derivative of a given function ( a 1D array) using Array. Here is the code
...
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
4
votes
2
answers
556
views
Chebyshev/Lagrange polynomials in spectral methods
I am currently trying to familiarise myself with (Pseudo-)Spectral Methods for solving differential equations. Now, I am struggling to understand some obviously crucial concept of this approach. The ...
- 185
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
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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
1
vote
1
answer
61
views
Applying spectral method to a damped, driven 2D bending-mode wave equation on an irregular domain with heterogeneous boundary conditions
I'm trying to model some two-dimensional waves, and am unsure how to combine my boundary conditions with spectral methods. The PDE I'd like to explore resembles the equations for damped, driven ...
- 153
3
votes
1
answer
202
views
orthogonal basis functions on arbitrary domains and boundary conditions
I'm interested in solving an inverse coefficient problem for a PDE.
Let's say the field to be estimated is $\theta$.
The conventional approach would be to use a finite element discretization for $\...
- 10.5k
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
0
votes
2
answers
559
views
Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?
I'm solving a local instability problem for a pipe Poiseuille flow.
The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial).
The basic flow is $\bar{u_r}=0, \bar{u_\...
- 1
2
votes
1
answer
456
views
What does the Chebyshev differentiation matrix look like for third and fourth derivative?
I have a PDE that contains both the 3rd derivative and 4th derivative. Example shown below
$$ \frac{\partial u}{\partial t} =\frac{\partial}{\partial x}(u^3\frac{\partial^3u}{\partial x^3}) $$
$$ \...
- 73
1
vote
0
answers
250
views
How to discretize PDEs using Chebyshev spectral method into a system of differential algebraic equations (DAEs)?
Let's take the heat equation. We have a time derivative and spatial derivative. How to discretize the spatial derivative using Chebyshev spectral method and convert it into DAEs? Like in the form of $ ...
- 73
3
votes
1
answer
836
views
Solving ODE with Spectral Method using Chebyshev Polynomials
I would like to solve following the basic equation of linear elasticity (for simplicity in 1D)
$$
\frac{d}{dx} \left( E \frac{du}{dx} \right) = 0 \quad \textrm{with} \quad u(1)=0, \; u(-1)=b
$$
...
- 33
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:
$$T-...
- 840
1
vote
1
answer
2k
views
How to solve heat equation in spherical coordinates with finite differences?
I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only.
...
- 111
1
vote
1
answer
369
views
Solving Poisson-like PDE with FFT
Problem
I have an $n\times n$ grid, and each point on the grid is assigned two values: a score, and an (inverse) speed factor. There is a "turtle" moving along the grid, and it's goal is to ...
- 131
3
votes
1
answer
128
views
Calculating the Jacobian for a function containing a derivative
I have the equation
$F(t) = \phi u + \frac{1}{2}\frac{d^2u}{dt^2} + u^3$
and broadly speaking, my task is to calculate the $\phi$ and $u(t)$ such that $F(t) = 0$.
I am testing out a new algorithm to ...
- 131
3
votes
2
answers
2k
views
Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods
Lately, I've been trying to solve numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods.
Let $\nu$ be the viscosity and $[0,L]$ the domain. The 1D equation is,
$$
u_t + uu_x + u_{xx} ...
- 173
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
1
vote
0
answers
62
views
Finite element method for an equation requiring switch between spectral and temporal domain
Some equations (such as the non-linear schrödinger equation for pulse propagation) are more easily solved in the spectral form, but still need a representation in the temporal domain to calculate ...
- 563
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
3
votes
1
answer
146
views
How does the diffusion of a finite volume method with a WENO scheme compare with that of spectral methods?
I know that, in general, finite volume (FV) methods are more (numerically) diffusive than spectral methods. However, I can't find any information on how the advection scheme changes that.
For example, ...
- 155
6
votes
1
answer
2k
views
Gauss-Lobatto quadrature and nodal points for FEM
By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.)
...
- 128
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:
$$
k =
\begin{cases}
1 ~\text{if} ~x &...
- 601
1
vote
2
answers
811
views
How to use the Thomas-Algorithm to the Heat-diffusion-equation correctly
My post is structured in four parts:
I give you some information about the context my principal questions refer to.
I will tell you what I believe to know about the Thomas Algorithm. If I am wrong ...
- 11
1
vote
2
answers
537
views
Solve wave equation with discontinuous coefficients numerically?
I would like to solve the following equation
$$\frac{\partial^2 y}{\partial t^2} - c^2(x,t)\frac{\partial^2 y}{\partial x^2}=0,$$
for $y=y(x,t)$ numerically. The wave speed, $c(x,t)$, is of the form
$$...
- 219
2
votes
1
answer
237
views
Pros of Fourier-Galerkin spectral methods
What are the pros of Fourier-Galerkin spectral methods while solving PDEs?
Here's the one that came in my mind first:
Easy implementation: using this method, differentiation operator computation is ...
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