Questions tagged [finite-differences]

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Ensuring symmetry in mixed derivatives using RBF-FD method

I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a bivariate function $f(x, y)$ at a set of ...
Rule's user avatar
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1 vote
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47 views

Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues

I am looking for a reference justifying the following statement. Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$ $$ ...
leo monsaingeon's user avatar
9 votes
2 answers
346 views

Change of variable formulas in discrete calculus?

Crossposted from MSE. In discrete calculus one defines the $h$-difference operator $$\Delta_h[f(x)] = f(x+h) - f(x)$$ and we often define $\Delta = \Delta_1.$ We can similarly define the indefinite ...
KJL's user avatar
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27 views

How to deal with discontinous problem with numerical method?

I would like to consider how to deal with the indicator function in a PDE. For example, for the PME with 𝑚=5, the initial condition is the two-Box solution with the same height, namely $$ u_0(x)= 1 \...
Chet's user avatar
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1 vote
1 answer
102 views

How to numerically solve differential equations involving sines, cosines and inverses of the unknown function? [closed]

Crossposted at SciComp SE I'm very new to finite difference method and I am just introduced to methods of solving differential equation using finite difference method via sparse matrix method. I find ...
Hari Sam's user avatar
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0 answers
24 views

Generating a proper finite difference scheme

I have recently started studying the finite difference schemes for numerical analysis. While I can now calculate difference schemes fairly easily for simple equations, I've recently come across a ...
Syed Ali Mohsin Bukhari's user avatar
2 votes
1 answer
107 views

Finite difference approximation

I'm trying to find formulas for the finite difference approximation "Five-points-stencil" of the first derivative for non-constant grid spacing. It's needed for the outermost left and right ...
Gogoman96 X's user avatar
4 votes
2 answers
424 views

What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation?

Let \begin{equation*} \begin{split} M_m &=\begin{pmatrix} -\binom{1}{0} & \binom{2}{0} &-\binom{3}{0} &\dotsm & (-1)^{m-1}\binom{m-1}{0} & (-1)^m\binom{m}{0}\\ 0 & \binom{2}...
qifeng618's user avatar
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8 votes
1 answer
679 views

Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials

Let $ X_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y_N = \{v =(v_0, \cdots, v_{N-1}): v_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X_N $ is the space of ...
Chushamm's user avatar
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2 votes
1 answer
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Algorithmically finding mixed-derivative coefficients from finite differences

This is a cross-post from Math.SE since the question has received very little attention. If it is too trivial, or in other ways not suited for this site, please let me know and I will remove it. ...
JCGoran's user avatar
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159 views

Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$. As an example, let $p(x)=x^{2}+2x$. Then ...
Mirar's user avatar
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2 votes
1 answer
53 views

High order difference operator applied to 1/u

I need a formula for $\Delta^k \frac 1 u$, where $u(x)$ is a strictly positive function, $\Delta^k$ is the difference operator defined recursively as $\Delta^k=\Delta^1 \Delta^{k-1}$ and $\Delta^1 u(x)...
Piero D'Ancona's user avatar
3 votes
0 answers
109 views

Smoothly connecting PDEs with finite differences

A PDE with non-smooth inhomogeneity Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$. I'm numerically solving the inhomogeneous PDE \begin{...
Alex's user avatar
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1 answer
183 views

Finding numerical solution for nonlinear Poisson-like equation using finite difference method

I am trying to use finite difference method to solve for $u(x,t)$ in the equation: \begin{align} \frac{\partial^2u}{\partial x^2} = \frac{au}{1+bu}, \end{align} which is actually part of a system of ...
mohd's user avatar
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0 answers
100 views

discrete Fourier transform for composition of differential operators on a grid

This question pertains to stability analysis of finite difference methods using the discrete Fourier transform. Suppose I have a convection diffusion equation of the form: (1) $\hspace{.5in}u_t + \...
GeauxMath's user avatar
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5 votes
1 answer
155 views

Finite difference for a highly nonlinear equation - The wind within the forest

Based on the Navier-Stokes equations and a few parameterizations, the horizontal steady-state wind $u(z)$ within a forest of height $H$ satisfies: $$ a\Big(\frac{du}{dz}\Big)^{\!2} + b\frac{du}{dz} \...
Matt's user avatar
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1 answer
260 views

Decomposing functions to Taylor-Fourier series

[Cross posted from Math.SE due to lack of attention] A great many functions can be expressed as a series of the form $$ U_0(x) + U_1(x) x + U_2(x) \frac{1}{2!}x(x-1) + ... $$ Where $U_r(x)$ are ...
Sidharth Ghoshal's user avatar
2 votes
0 answers
79 views

discrete parabolic Harnack inequality

I am currently looking for a discrete version of the parabolic Harnack inequality in the following "$L^1$ to $L^\infty$" form: If $u(t,x)\geq 0$ is a (say, smooth) subsolution of \begin{equation} ...
leo monsaingeon's user avatar
1 vote
1 answer
191 views

Finite differences of Stirling numbers

Let s(n,k) and S(n,k) denote the Stirling numbers of the first (with signs) and second kinds, respectively. Next consider the sequence |s(n+2,n)| which begins: (2,11,35,85,175,...) . Using this to ...
user2052's user avatar
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1 vote
1 answer
383 views

Laplace equation, medium discontinuity and finite difference method

The main question is: How to deal with the Poisson equation in the presence of the medium interface. Let's say we have 1D Laplace equation: \begin{equation} -\frac{d}{dx}\left(\epsilon(x)\frac{d}{dx}\...
drszdrsz's user avatar
1 vote
0 answers
33 views

How to change difference equation time steps when rearranging?

I am using difference equations to solve SDOF systems. I have the system $$m\ddot{y_i}+c\dot{y_i} + ky_i = x_i$$ Using the difference equation results for the derivatives, I am meant to end up with ...
Joshua Jones's user avatar
6 votes
2 answers
479 views

kth finite difference always positive when kth derivative is?

Let $f : \mathbb{R} \to \mathbb{R}$ such that the $k^{\rm th}$ derivative of $f$ is strictly positive for every $x \in \mathbb{R}$. Define the forward difference operator to be: $$\Delta(g,h) = g(x+h)...
George Shakan's user avatar
8 votes
1 answer
3k views

Review paper/book on Finite Difference Methods for PDEs

I am looking for a good, relatively modern, review paper/book on Finite Difference Methods for PDEs with a theoretical emphasis in mind. By theoretical emphasis I mean that I care about theorems (i.e. ...
Lentes's user avatar
  • 391
4 votes
0 answers
684 views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well. I am reading a paper [1] where they solve the following non-...
Hunter's user avatar
  • 215
4 votes
1 answer
2k views

Numerically calculating the divergence of a set of oriented points

Say I have a set of oriented points at locations $\vec{v_i}$ with each some direction $\vec{n_i}$, in practice they represent a surface with its normals. How would I calculate the divergence of this ...
Jan M.'s user avatar
  • 143
5 votes
1 answer
459 views

Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
Sridhar Ramesh's user avatar
1 vote
0 answers
503 views

9-point stencil "equivalent" for advection equation [closed]

So I inherited from some people a code that solves the advection-diffusion-reaction equation for a particular system. The original code was first implemented in 1D which worked fine in cartesian ...
mathdummy's user avatar
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4 votes
2 answers
787 views

Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...
2 votes
0 answers
152 views

probabilistic interpretation of a finite difference scheme

Let me start with some simple background. Consider the heat equation : $ \frac{\partial p}{ \partial t} = \frac{1}{2} \frac{\partial^2 p}{\partial y^2} \quad \mbox{in} \quad \mathbb{R}\times (0,\...
megaproba's user avatar
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0 answers
37 views

Any reference in absorbing boundary conditions for non-abelian gauge fields?

Is there any paper on absorbing boundary conditions for non-abelian gauge fields? Currently I only saw some on elastic wave equations and some on EM fields.
Idear's user avatar
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0 answers
226 views

GKS stability of a finite difference scheme

In this paper, I can not reproduce the results obtained equation 62. I have tried to reproduce it using Wolfram alpha but the results are different. However, using equation (40) instead of the one ...
user1824346's user avatar
3 votes
1 answer
811 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ D_{j}f(x):=...
GFF41's user avatar
  • 31
7 votes
2 answers
8k views

Conditions for convergence of Euler's method

It is known that a sufficient and necessary condition for $$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$ to have a unique solution is $f$ Lipschitz in $y$ and continuous in $t$. However, ...
John Wong's user avatar
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1 vote
1 answer
200 views

Uniqueness of Newton (modulo a constant) series on a compact set

Good morning everybody. My question is as follows: Let $K$ be a compact subset of $\mathbb R$ and let $v\in C^\infty(K)$. Consider the finite difference operator $\Delta v(x)\doteq v(x+1)-v(x)$. It ...
user avatar
2 votes
2 answers
584 views

What summations of elementary trig functions are known to have (elementary) closed forms?

I've been trying to find a closed form of $\displaystyle \sum_k{\tan{(k)}}$ that contains only elementary functions, and I think I may be onto something. But rather than reinvent the wheel, I want to ...
Matt Groff's user avatar
3 votes
1 answer
85 views

Finite differencing scheme for Hamilton's equation with planar linkages

I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and $${{dp}\...
Charles Baker's user avatar
4 votes
2 answers
468 views

Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

I would like to numerically solve a hyperbolic PDE of the form $\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...
Michael Andrew Bentley's user avatar
0 votes
0 answers
246 views

Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per vertex)....
paghdv's user avatar
  • 101
0 votes
1 answer
298 views

Limits of functions with converging zeros

What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros? More precisely, suppose that $f_i: R^n \to R^m$ is a ...
Chris Woodward's user avatar
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0 answers
67 views

Proof that Newton expansion over derivatives has the properties of an integral [duplicate]

Let's consider a Newton expansion over consecutive derivatives of a function: $$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ Can it be proven that such ...
Anixx's user avatar
  • 9,366
0 votes
1 answer
161 views

Are all discrete-analytic funtions as defined here also natural?

Let's define a discrete-analytic function as a function that is equal to its Newton expansion: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=...
Anixx's user avatar
  • 9,366
7 votes
0 answers
336 views

Polynomials and divided differences

I would greatly appreciate any hint for proving the following. Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then $...
George's user avatar
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2 votes
1 answer
103 views

Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives to solve this IVP problem ?

I want to numerically integrate the equation $\partial_t u= a(t) \partial_xu+b\partial_{xxx}u+c$ to get $u(t)$. Is is preferable to use a difference formula of higher order of accuracy for spatial ...
gstar2002's user avatar
2 votes
0 answers
838 views

Problem using finite difference to solve a initial value problem

I tried to use 'finite difference' method to solve an Initial Value Problem (IVP). For the two boundaries I used periodical condition and for the differential operators I used 4th degree center ...
gstar2002's user avatar
4 votes
2 answers
1k views

Convergence of Newton series for sin ax

Let's define half discrete-analytic function as a function whose Newton series converges to that function for each $x>0$: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\...
Anixx's user avatar
  • 9,366
14 votes
3 answers
1k views

How many sequences of rational squares are there, all of whose differences are also rational squares?

After commenting on a question of Joseph O'Rourke's, I thought it interesting that a number theory result (artihmetic progressions of rational squares cannot be arbitrarily long) had applications to ...
Gerhard Paseman's user avatar
7 votes
0 answers
1k views

Are these two functions equal?

The question here is sparked by the discussion inside this question about indefinite sum(antidifference) of tan(x). A proposed solution was a function $$f_1(x)=ix-\psi _{e^{2 i}}^{(0)}\left(x+\frac{\...
Anixx's user avatar
  • 9,366
20 votes
9 answers
10k views

What is the indefinite sum of tan(x)?

What is the indefinite sum of the tangent function, that is, the function $T$ for which $\Delta_x T = T(x + 1) - T(x) = \tan(x)$ Of course, there are infinitely many answers, who all differ by a ...
Herman Tulleken's user avatar