A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.
Questions tagged [finite-differences]
706 questions
107
votes
3 answers
Is the derivative the natural logarithm of the left-shift?
(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.)
I noticed something really neat the other day.
Suppose we…
David Zhang
- 8,302
- 2
- 35
- 56
29
votes
2 answers
Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$
After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form
$\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ is a constant and $\epsilon_k=\pm1$. I discovered…
Pauly B
- 5,162
- 1
- 12
- 23
28
votes
2 answers
Has anybody ever considered "full derivative"?
When differentiating we usually take a limit and drop the infinitesimal terms.
But what if not to drop anything?
First, we extend the real numbers with an infinitesimal element $\varepsilon$ which has its own inverse $1/\varepsilon=\omega$.
And…
Anixx
- 7,543
- 1
- 25
- 46
23
votes
7 answers
What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs?
Can you help me explain the basic difference between FDM, FEM and FVM?
What is the best method and why?
Advantage and disadvantage of them?
Anh-Thi DINH
- 463
- 1
- 6
- 15
17
votes
4 answers
Why isn't finite calculus more popular?
I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful tool for evaluating sums, essentially a…
Elliot Gorokhovsky
- 1,888
- 1
- 16
- 28
16
votes
5 answers
Chain rule for discrete/finite calculus
In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in simplified or otherwise helpful terms?
It's probably not…
GregRos
- 1,727
- 13
- 27
10
votes
2 answers
Impose PDE itself as Boundary Condition?
Consider, for example, the elliptic PDE $u_{x}+u_y+u_{xx}+u_{yy}=0$ for $(x,y)\in[0,\infty)^2$.
Solution methods often require me to impose boundary conditions. Often, these arise naturally from applications (physics, biology, economics etc.). But…
Alex
- 629
- 2
- 18
9
votes
3 answers
Is there a mean value theorem for higher order differences?
The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if there is a similar formula for higher order…
nullUser
- 26,284
- 5
- 66
- 124
8
votes
0 answers
Convergence of numerical methods for Viscous Burgers' Equation
For linear problems we can deduce convergence of numerical schemes by checking consistency and stability because of the Lax Equivalence Theorem.
For conservation laws, we know that conservative, consistent and monotone schemes will converge to the…
428
- 525
- 4
- 11
7
votes
1 answer
What is convection-dominated pde problems?
Can you explain for me what is convection-dominated problems? Definition and examples if possible.
Why don't we can apply standard discretization methods (finite difference, finite element, finite volume methods) for convection-dominated…
Anh-Thi DINH
- 463
- 1
- 6
- 15
7
votes
3 answers
Finite differences second derivative as successive application of the first derivative
The finite difference expressions for the first, second and higher derivatives in the first, second or higher order of accuracy can be easily derived from Taylor's expansions. But, numerically, the successive application of the first derivative, in…
CFDIAC
- 73
- 1
- 4
7
votes
2 answers
Recurrence relations on a continuous domain
While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, t_n$ satisfies
$$ N(t) = N(t-t_1) + N(t-t_2) + \dots…
ShreevatsaR
- 39,794
- 7
- 90
- 125
6
votes
1 answer
9 point stencil for Laplacian operator
Given the following 9 point Laplacian
\begin{align}
-\nabla^2u_{i,j} = \frac{2}{3h^2}\left[5u_{i,j} - u_{i-1,j} - u_{i+1,j} - u_{i,j-1} - u_{i,j+1} - u_{i-1,j-1} - u_{i-1,j+1} - u_{i+1,j-1} - u_{i+1,j+1}\right] \quad\quad (1)
\end{align}
Show using…
James
- 61
- 1
- 2
6
votes
2 answers
Show that the $k$th forward difference of $x^n$ is divisible by $k!$
Define the forward difference operator
$$\Delta f(x) = f(x+1) - f(x)$$
I believe that if $f(x)$ is a polynomial with integer coefficients, $\Delta^k f(x)$ is divisible by k!. By linearity it suffices to consider a single monomial $f(x) = x^n$. …
Geoffrey Irving
- 796
- 4
- 15
6
votes
0 answers
What is a common framework for these divergent sums?
If you expand $2^x$ using a finite difference series you end up with the formula
$$ 1 + x + \frac{1}{2!}x(x-1) + \frac{1}{3!}x(x-1)(x-2) ... = \sum_{n=0}^{\infty} \frac{(x)_n}{n!} $$
Now these series diverge for negative arguments, but they give…
Sidharth Ghoshal
- 14,653
- 8
- 33
- 81