All Questions
41 questions with no upvoted or accepted answers
6
votes
0
answers
709
views
Stochastic Galerkin projection approach for using generalized polynomial chaos expansion (GPCE) in solving PDE
I want to know if there is any way to define the test and trial function in the way that I want instead of using the default functions. So if I want define the polynomial and basis and coefficient, ...
- 111
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
75
views
Spectral solver on em-pic
I'm recently studying for the spectral solver to implement EM-PIC code. I read an article and have some questions.
Many PIC codes uses spectral solver to overcome numerical artifacts on FDTD.
In the ...
- 49
3
votes
0
answers
243
views
how to construct chebyshev differentiation matrix of even/odd polynomial
I know there is efficient way to construct chebyshev differentiation matrix $D_N$. The idea is to fit the function $f(x)$ with a polynomial of $p_N(x)$ of order $N$, and use the derivative $p_N'(x)$ ...
- 31
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
3
votes
0
answers
144
views
Closed form PDF/CDF using Orthogonal Polynomial Expansion (gPC)
Consider a random variable which is given by an orthogonal polynomial expansion in one parameter (or polynomial chaos expansion PCE), i.e. ,
$$ f(\alpha) = \sum\limits_{n=0}^{\infty} \hat{f} (n) \psi ...
- 245
3
votes
0
answers
96
views
Differential eigenproblem with eigenvalue in boundary condition
Statement of the problem
I need to (numerically) solve an eigenproblem of the type
$$-\omega^2\mathcal{D}_1\vec{x}=\mathcal{D}_2\vec{x}$$
on the interval $[-1,1]$, where $\mathcal{D}_1$ and $\...
- 131
3
votes
0
answers
137
views
Spectral/hp-finite elements for 4th order PDEs
Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, ...
- 188
3
votes
0
answers
249
views
Spectral Collocation (or Weighted Residual) Methods to solve Stiff ODEs?
I have a system of ODEs which is (at least moderately) stiff.
Consider the class of spectral collocation methods https://en.wikipedia.org/wiki/Spectral_method or the related class of weighted ...
- 376
3
votes
0
answers
198
views
Integration of nonlinear PIDE via spectral methods
At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\...
- 203
3
votes
0
answers
274
views
Choosing good basis functions to approximate a Lipschitz function
Let $D = \left\{0, t_1, t_2, \ldots, t_n\right\} \times [0,1]$
and
$$
f: D\to [0,1],
$$ be a function of time and a one-dimensional space.
There is no analytical formula for $f$, but $f(t_i, \cdot)$ ...
user729
2
votes
0
answers
82
views
Preventing an Overflow in Exponential Integrating Factor
The following is the Discrete Spectral Vorticity Evolution PDE for incompressible flow:
$$ \frac{\partial \Omega_{pq}}{\partial t} = \nu \left( \frac{\partial^2 \Omega_{pq}}{\partial x^2} + \frac{\...
- 193
2
votes
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
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
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
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:
$$T-...
- 840
1
vote
1
answer
256
views
Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\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
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
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
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
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
1
vote
0
answers
240
views
Poisson equation with FFT and normalization
I'm trying to understand how to solve Poisson equation with FFT. Say, if we have the simplest periodic example
$$u_{xx}=-4\pi\cos(x)$$
The solution then should be
$$u=4\pi\cos(x)$$
I really get ...
- 11
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
1
vote
0
answers
953
views
Methods and tools to solve the two-temperature model (TTM)
I would like to model heat diffusion at the gold / water interface after excitation of the metal surface by an ultrafast laser pulse (ca. 80 fs).
An appropriate model to start with would be the "two ...
- 21
1
vote
0
answers
137
views
Solve a PDE BVP with Spectral methods in time and space?
I have a PDE (coming from a Bellman equation with $z$ under brownian motion). Let $z \in [0,\infty)$ and $t \in [0,T]$. To sketch the equation:
$$
(r - g(t)) v(t,z) = \pi(t,z) + (\gamma - g(t))v_z(...
- 376
1
vote
0
answers
212
views
BTCS-like method for heat conduction in unstructured triangular grid
I want to write a simple simulation for heat conduction in a unstructured triangular mesh.
I already made it work for a structured rectangular grid with the ADI method, but now I need more complex ...
- 23
1
vote
0
answers
98
views
Find B-spline coefficients from values on collocation points
The context of my question is how to compute high order derivates on direct numerical simulation of turbulent channel flow. It is of particular interest for fluid dynamics and turbulence research.
I ...
- 121
1
vote
0
answers
74
views
Pseudo-Spectral cosine transform
I'm trying to solve the following equation
$$u_t = u_{xx} + u(1-u^2), u_x(\pm 1) = 0,$$
using the Fourier cosine transform. The nonlinear term gives a convolution which I would rather avoid, which is ...
- 184
1
vote
0
answers
84
views
Perfect filtering of high frequencies in 2D FFT (Multidimensional 2/3 Rule)
Let $u_n$ be an array containing discrete values of the function $u(x,y)$. Performing a 2D FFT to this array we obtain $\hat{u_n}$ representing the values of $\hat{u}(k_x,k_y)$. I would like to ...
- 113
1
vote
0
answers
66
views
Second and Higher Order Order Corrector in Spectral Deferred Correction
I am trying to work out a second order or higher order correction for the method of Spectral deferred Correction (SDC). Specifically using as a corrector a second order or third order multi-step.
In ...
- 11
1
vote
0
answers
144
views
Implicit time integrator for Chebyshev collocation method for linear hyperbolic system
I want to solve linear hyperbolic system using Chebyshev collocation method. As this method puts severe constraint on the time step for the explicit time integration, I decided to switch to implicit ...
- 508
1
vote
0
answers
43
views
Aliasing question for a spectrally derived Lagrangian
Consider two functions
$$X=\sum_{n=-N}^N i\ sign(n) y_n e^{-in\xi}; \quad \quad Y = \sum_{n=-N}^N y_n e^{-in\xi}.$$
where $y_n$ are the dependent variables of the system, and are functions of time. ...
- 385
1
vote
0
answers
345
views
Am I using the incorrect implementation of the fast Chebyshev transform?
I was told that the fast Chebyshev transform has superior spectral convergence, but I am unable to verify its rumored convergence. I was given plots of its spectral convergence, where the signal's ...
- 113
0
votes
0
answers
128
views
Heat Equation for fast source with FiPy
I'm trying to solve the following differential equation with FiPy, basically laser irradiation on a surface
$$
\rho_{s}C_{p,s}\frac{\partial T}{\partial t} = k_{s}\frac{\partial^{2}T}{\partial x^{2}} +...
- 11
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
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
0
votes
0
answers
64
views
Spectral Clustering by Andrew Ng paper: theorem proof question
I recently read the paper "On Spectral Clustering: Analysis and an Algorithm" by Ng et al.
Much of the paper centers on Theorem 2 and equation 8.
To me, it appears there is no given or referenced ...
- 11
0
votes
0
answers
300
views
How to form the stiffness matrix for the Poisson equation using a spectral method
This is a follow-up question to How do I form the Chebyshev differentiation matrix in MATLAB? The goal of the following code is to solve the Poisson problem:
...
user21030