The process of representing a continuum space with a finite set of points/elements

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3
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2answers
96 views

2nd order centered finite-difference approximation of $u_{xy}$

The problem is to find a 2nd order finite difference approximation of the partial derivative uxy, where u is a function of x and y. Page 5 of this pdf I found does a centered difference approximation ...
4
votes
1answer
122 views

Which finite-element framework allows the easy implementation of HDG methods?

I have tinkered around with the implementation of hybridizable Discontinuous Galerkin (HDG) methods in DUNE-PDELab and DUNE-FEM for my PhD but since then I have not had the time to work on these codes ...
0
votes
0answers
28 views

Discretization using compact finite differences

I am having difficulty discretizing an equation using compact finite differences. My understanding of compact finite differences is that instead of having an explicit equation for the derivative, ...
4
votes
2answers
92 views

Discretization method for a reaction dominated elliptic PDE

I'm working with an elliptic reaction diffusion PDE of the form $$-k\nabla^2u+cu=0$$ I've noticed that when the reaction term dominates over the diffusion (i.e. $c>>k$), the true (exact) ...
0
votes
0answers
101 views

discretizing $\frac{d^2}{dx^2}$

I am trying to discretize the differential operator $\frac{d^2}{dx^2}$ acting on $S^1 = [0,1]$ using finitely many points around a circle at $0, \frac{1}{N}, \frac{2}{N}, \dots, \frac{N-1}{N}$. Here ...
6
votes
2answers
144 views

Spectral Methods in time

I was reading up on Spectral Methods for PDEs. In all the descriptions I read, while the position component is approximated via a Fourier series or other methods, the time component is still ...
6
votes
1answer
168 views

When is discrete Fourier transform a good approximation to the continuous one?

I am looking for help in understanding the use of discrete Fourier transform as an approximation to continuous Fourier transforms. As an exercise, I considered a Gaussian $$f(x) = ...
5
votes
2answers
210 views

Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) finite volume method

I have been trying to debug this error the last few days I wondered if anybody has advice on how to proceed. I am solving the Poisson equation for a step charge distribution (a common problem in ...
3
votes
3answers
105 views

Pseudo-inverse of a discretized operator with a null space?

Is there a way to understand what happens when a singular operator is discretized and inverted using the pseudoinverse (say using the SVD Moore-Penrose pseudoinverse)? For example, if we discretize ...
3
votes
1answer
115 views

Should the Jacobian of a system of PDEs be calculated from the main equations of the discretised equation?

I am solving a coupled system of non-linear PDEs in 1D. Something like, $$ u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w) $$ where each variable is a function of $x$ (the spatial ...
2
votes
2answers
1k views

Time stepping in comsol multiphysics

I have an important question I wasn't able to find an answer to. i would like to know which is the algorithm comsol uses in order to correct the time step it uses. For example, when you try to solve ...
5
votes
1answer
171 views

Implementing an adaptive discretisation (upwind/central hybrid) for the advection-diffusion equation on a non-uniform mesh

I am following the approach of Hundsdorfer from Numerical solution of time-dependent advection-diffusion-reaction equations in which they introduce adaptive upwinding. The method can adapt from pure ...
9
votes
3answers
231 views

How should non-constant coefficients be treated with finite-volume first order upwind scheme?

Starting with the advection equation in conservation form. $$ u_t = (a(x)u)_x $$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...
4
votes
2answers
131 views

Improving the time integration of implicit discretized PDE with a non-linear source term

This might be a naive question, but when applying a implicit discretization to a PDE with a source term, should the source be averaged in time? For example if we take the diffusion equation with a ...
5
votes
1answer
180 views

Boundary value method for equation $u_{tt} = u_{xxx}$

I have this funny equation $$ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^3 u}{\partial x^3}, \qquad x \in [0,1], \qquad t \in (0,T] $$ with initial conditions $u(x,0) = \sin(2\pi x)$, ...
2
votes
1answer
142 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 ...
11
votes
2answers
214 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 $0 < \varepsilon \ll 1$ ...
4
votes
1answer
256 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
456 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
95 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
99 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
188 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 ...
4
votes
1answer
131 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
42 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} + ...
14
votes
2answers
1k 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
57 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
29 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?
1
vote
1answer
269 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
173 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 ...
16
votes
3answers
391 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? ...
3
votes
1answer
153 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 ...
6
votes
2answers
1k 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
406 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 ...
8
votes
1answer
107 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 ...
3
votes
1answer
457 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 ...
13
votes
3answers
1k 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} - ...
7
votes
2answers
152 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
161 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 ...