The process of representing a continuum space with a finite set of points/elements
2
votes
0answers
41 views
Explicit 4th order space wave equation not stable implementation?
The explicit 4th order discretization for the 2D scalar wave equation is given by:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} }{\Delta s} \right) ^2 \left( \sum_{a=-N}^N w_a ...
9
votes
2answers
109 views
Oscillations in singularly perturbed reaction-diffusion problems with finite elements
When FEM-discretizing and solving a reaction-diffusion problem, e.g.,
$$
- \varepsilon \Delta u + u = 1 \text{ on } \Omega\\
u = 0 \text{ on } \partial\Omega
$$
with $0 < \varepsilon \ll 1$ ...
3
votes
0answers
59 views
Von Newman stability analysis for 2D acoustic wave equation explicit
Von Newman stability analysis for acoustic wave equation explicit centered differences: 2nd order time and space (N 2)'th order:
\begin{eqnarray}
U_{jk}^{n+1} = \left( \frac{\Delta t V_{jk} ...
1
vote
1answer
52 views
Discrete 3D convolution of matrix valued functions
As stated in the title, I need to make a discrete 3D convolution of two matrix-valued functions. Or, actually, I need to convolve a matrix-valued function with a vector-valued function. That is, I ...
2
votes
1answer
45 views
Flux calculation - discretization of solid angle
I am currently tasked with calculating the total flux of photons or irradiance from a flat emitter ('pixel'). Previously we measured the Luminance head-on (90 degree from the emitting surface) and ...
1
vote
1answer
51 views
How can I quantify the error of FFT-based poisson solvers?
I have an FFT code that solves a particular case of the steady Euler equations where a Poisson equation is solved, what is a good way to quantify the error? Is what I am doing ok?
Since I do not have ...
1
vote
1answer
62 views
Von Neumann Stability Analysis
I came across the following task recently:
Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
3
votes
1answer
71 views
Mehrstellenverfahren for Poisson?
I found this method in this book for solving the Poisson equation with error converging with $\mathcal{O}(h^4)$. However, when I try to implement it for the 1D equation $-u''(x)=f(x)$, it only ...
0
votes
1answer
33 views
Clarification on interpolation equalities given by Briggs
Briggs, "A Multigrid Tutorial" (pg. 35) has the following expressed as 2-D interpolation:
\begin{align*}
v^h_{2i,2j} &= v_{i,j}^{2h}\\
v^h_{2i+1,2j} &= 0.5\cdot(v_{i,j}^{2h} + ...
10
votes
2answers
257 views
Is Crank-Nicolson a stable discretization scheme for Reaction-Diffusion-Advection (convection) equation?
I am not very familiar with the common discretization schemes for PDEs. I know that Crank-Nicolson is popular scheme for discretizing the diffusion equation. Is also a good choice for the advection ...
2
votes
0answers
52 views
What are Implications of Commutative Diagrams?
This question may be too broad. But I really want to know some concrete explanations. I often find various commutations appear here and there, which concerns the application order of two operators. ...
2
votes
0answers
23 views
Lax equivalence theorem for integro-differential equation
Can the Lax equivalence theorem (http://en.wikipedia.org/wiki/Lax_equivalence_theorem) be applied to the discretization of integro-differential equations, or does a similar theorem exist for them?
2
votes
1answer
153 views
How to apply a Galerkin finite element method to a linear, one-dimensional boundary value problem
I have the following boundary value problem:
$$-(\alpha u')' + \gamma u = f $$
in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$
and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re $
The weak ...
5
votes
1answer
145 views
How far is a non-symmetric discretization of an elliptic operator from the continuous operator itself?
I am investigating the accuracy and stability properties of a non-symmetric discretization of a Poisson problem. The problem originates from a ghost fluid discretization of the projection step of a ...
15
votes
3answers
300 views
Why is the time dimension special?
Generally speaking, I've heard numerical analysts utter the opinion that
"Of course, mathematically speaking, time is just another dimension, but still, time is special"
How to justify this? ...
2
votes
1answer
137 views
Which discretization scheme to use for elliptic PDE?
While simulating motion of nonlinear inelastic wire one meets the following equations
\begin{align}
&{\partial^2\varphi\over\partial t^2}=2{\partial F\over\partial s}{\partial\varphi\over\partial ...
5
votes
2answers
598 views
structured grid and unstructured grid
I am new to the field of CFD. When should one go for structured grid and when should one go for unstructured?
(Yes, it depends a lot on the geometry of the problem)
More specifically, I want to know ...
4
votes
1answer
241 views
Python syntax for MATLAB/Octave colon operator a:dx:b
I am trying to rewrite some MATLAB/Octave code in Python, and I don't know what would be the nicest or most intuitive way of writing
...
7
votes
1answer
85 views
Eigenspace basis continuously depending on parameters
I have a Hermitian matrix $\mathbf{H}$ which depends on two parameters say $x$ and $y$.
When I diagonalize it at two close points $(x_1,y_1)$ and $(x_2,y_2)$ I get two close eigenvalues ...
1
vote
1answer
300 views
How to obtain an implicit finite difference scheme for the wave equation?
Suppose I had the following problem:
$U_{tt}=U_{xx}+U_{yy}$ in $\Omega=[0,1]\times[0,1]$
$U(x,y,0)=f(x,y)$
$U_{t}(x,y,0)=g(x,y)$
$U=0$ on $\partial \Omega$
I know that there is an explicit ...
12
votes
3answers
578 views
How can I numerically differentiate an unevenly sampled function?
Standard finite difference formulas are usable to numerically compute a derivative under the expectation that you have function values $f(x_k)$ at evenly spaced points, so that $h \equiv x_{k+1} - ...
6
votes
2answers
142 views
Is there one general approach to build a projection methods for different problems?
My question is probably going to be too general to answer it with a couple words. Could you please suggest a good reading in that case. Projection methods are used to reduce size of the solution space ...
4
votes
1answer
145 views
Discretization of Classical Density Functional Theory (CDFT) problems
I formulate questions about ion densities in biological and materials problems using Classical Density Functional Theory, as in this paper which is also on the arXiv. The discretization in that paper ...
