All Questions
Tagged with finite-difference discretization
49
questions
1
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0
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66
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Deriving order of accuracy and interpreting a given discretization scheme when underlying method ( finite difference/volume) not known
If a spatial grid is given with time levels like this:
to solve the following model problem
Now consider the following discretization schemes:
Scheme 1
Scheme 2
Usually, to determine order of ...
- 445
1
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0
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126
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A staggered grid for an eigenvalue problem (linear stability analysis)
I'm interested in extending the concept of a staggered grid (commonly used to solve the incompressible Navier-Stokes equations) to a linear stability analysis context. For example, we can consider ...
- 11
0
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1
answer
97
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finding discretization error in Burger equation
I was reading the paper given in the link http://www.unige.ch/~hairer/preprints/parareal.pdf and I have a problem in understanding in page 10 for the Burger equation on implementing Parareal method ...
- 41
1
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1
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262
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constructing a symmetric matrix for finite difference
I come across the following operator in a paper
$\mathcal{I}\psi = \psi_{xxxx} + (r~\psi_x)_x$,
where $\psi=\psi(x)$ and $r=r(x)$. Periodic boundary condition is employed. It claims that the operator $...
- 217
0
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1
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145
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Time & Space matlab discretization Finite Differences confusion
I have been trying to solve this equation and write the finite difference scheme in matlab for months, but I still am not successful.
Given the KdV Equation $$\tag{1}u_{t} -6uu_x+u_{xxx}=0$$
I have ...
- 23
1
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0
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62
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Good non oscilliatory derivatives for an exsisting grid
I'm calculating the entropy production of a shockwave by utilizing the equations:
\begin{equation}
\sigma = J'_q\frac{\partial}{\partial x}\left(\frac{1}{T}\right) +\frac{1}{T}\frac{4\eta}{3}\left(\...
- 55
1
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0
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425
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How to compute the Eigenvalue and Eigenstates of Quantum well with Effective mass using finite difference method in Python?
I want to compute the eigenvalues and eigenstates of a quantum well with different effective masses of electron in the barrier and in the quantum well. As can be seen [1]: https://github.com/mholtrop/...
- 21
2
votes
1
answer
832
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Tensor product representation for the 9-point finite difference approximations for the Poisson equation
If we use 5-point finite difference approximations in a uniform rectangular grid to solve the Poisson PDE
\begin{align}
-\Delta u &= f \ \ \text{en} \ \ (0,1)\times (0,1) \label{P1} \\
u &= ...
- 83
0
votes
1
answer
216
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Discretization of a non-linear ODE using FDM isn't grid indepenent
I am trying to solve the ODE :
$\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$
+
using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
1
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0
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69
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discretization of advection diffusion with variable coefficients
I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes
$$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
- 11
2
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0
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57
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In sights into why higher order finite differencing method leads faster to instability
I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
- 470
1
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0
answers
62
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Can the standard multigrid performance be used for time-dependent PDEs?
Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...
- 481
1
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0
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181
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FDM discretization of equation on the boundary
In order to simulate the following equation using FDM
$$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$
$$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$
$$(u_t(t,x)+u_{x}(t,x))\...
- 111
1
vote
1
answer
420
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Approximation Error in a Finite Difference Approximation of the Square of Derivative
First Part: (First-order derivative)
Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is
\begin{align}\tag{1}
f(x + h) = f(x) + hf'(x) + \frac{1}...
- 95
0
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1
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2k
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Discretization Neumann boundary condition
I'm currently working with the following Poisson equation with mixed boundary conditions, including a Neumann boundary condition.
$$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<...
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