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 ...
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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 ...
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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 ...
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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 ...
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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}...
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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 ...
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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. ...
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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 ...
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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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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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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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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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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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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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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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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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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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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 ...
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{\...
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 ...