Questions tagged [finite-differences]

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Existence of a solution for the discrete pde $\nabla a \nabla f = g$ in $\mathbb{Z}^d$ with $g$ periodic

I first asked this question on math.stackexchange as I first thought it was somewhat innocent. I would like if one can solve the problem $\nabla^* a \nabla f =g$ in $\mathbb{Z^d}$ with $f(0)=c$ where $...
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66 views

ODE operator splitting with second order time discretization not possible?

I am trying to solve an ordinary differential equation (ODE) using an operator splitting approach: $\frac{\partial f}{\partial t} = A(f) + B(f)$ Let's assume that $A$ and $B$ are very simple: $\...
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0answers
47 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 + \...
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1answer
128 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} \...
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1answer
199 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 ...
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70 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} ...
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1answer
107 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 ...
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1answer
185 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}\...
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15 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 ...
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2answers
395 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)...
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1answer
2k 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. ...
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576 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-...
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1answer
1k 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 ...
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1answer
325 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 ...
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465 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 ...
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2answers
742 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 ...
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0answers
132 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,\...
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33 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.
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164 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 ...
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1answer
504 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):=...
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2answers
3k 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, ...
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1answer
170 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 ...
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2answers
389 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 ...
2
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1answer
74 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}\...
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2answers
382 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 \...
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0answers
215 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)....
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1answer
241 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 ...
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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 ...
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1answer
144 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=...
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324 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 $...
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1answer
97 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 ...
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0answers
772 views

Problem using finite difference to solve a initial value problem

Hallo, I tried to use 'finite difference' method to solve a Initial Value Problem(IVP). For the two boundaries I used periodical condtion and for the differential operators I used 4th degree center ...
4
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2answers
808 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}^{\...
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3answers
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 ...
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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{\...
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9answers
9k 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 ...