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.

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Finite Difference for Hamilton-Jacobi-Bellman without boundary conditions

Let $t\in\mathbb{R}_+$ denote time, $x \in X$ is the state and $u \in U$ the control. The objective function is $F:X \times U \to\mathbb{R}$ and $f:X \times U \to\mathbb{R}$ is the law of motion for ...
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11 views

finite volume methods: what do I have to do with the cell averages after each step?

I'm having a hard time understanding finite volume methods. If I take for example the scalar advection equation $$\partial{u}_{t}+a\partial{u}_{x}=0, a>0$$ with suitable initial and bondary ...
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14 views

Reference request for finite difference method

I am trying to use finite difference method to solve the minimizing problem $$ J[u]=\min_{u\in BV(Q)}\{\|u-f\|_{L^1(Q)}+|u|_{BV(Q)}\} $$ where $Q=(0,1)\times (0,1)$ is a uint square and ...
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10 views

Group velocity navier-Stokes in finite difference

Implementing the linearized Navier-stokes equations for acoustic wave propagation in an homogeneous friction-less medium with background flow, that is, $$ \partial_t p = -\mathbf{w}.\nabla p - \rho ...
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20 views

calculus of finite differences

If D,E,$\delta,\mu$ be the operators with usual meaning and if hD=U, where h is the interval of differencing,How to prove the following relations between operators:- ...
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10 views

Optimal relaxation parameter for the SOR method when solving Poisson equation for non-square problems

Yang and Gobbert in paper "The Optimal Relaxation Parameter for the SOR Method Applied to a Classical Model Problem" ( http://userpages.umbc.edu/~gobbert/papers/YangGobbert2007SOR.pdf ) gave proof ...
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23 views

Numerical Differentiation

Determine the constants a,b,c and d such that the interpolating polynomial $y_x=y(x_0+sh)=ay_0(x_0-h)+by_1(x_0+h)+...$ Solution We have ...
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22 views

Proof of the Lax-Wendroff theorem

The Lax-Wendroff theorem says that, if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution. In the book "Numerical ...
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1answer
26 views

The effect of the CFL number in the numerical solution in this conservation law

I've been studying the very basics of numerical methods applied to conservation laws, and I'm having trouble understanding the role of the CFL number in the upwind scheme. I want to understand it (if ...
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144 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
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39 views

Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
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20 views

Finite differences on a hexagonal/triangular lattice with Cartesian coordinates

So, I've been thinking recently about how to approximate the Laplacian operator using finite differences on a non-square lattice. For example, on a typical square lattice, in a Cartesian coordinate ...
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14 views

Turning a difference equation in two unknowns into an eigenvalue problem

We were given a differential equation to solve numerically, {y''-3y'+2k^2*y=0, y(0)=y(1)=0}. We used the finite difference approximations to convert it into a diference equation. The following ...
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1answer
31 views

Unsure of 2d finite difference method with second order term?

I have the following equation for the price of Black Scholes Euro option - (1) $$-\frac{\delta C}{\delta t} = -\alpha\frac{\delta^2 C}{\delta x^2} + [r - \delta + \frac{\sigma^2}{2}]\frac{\delta ...
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1answer
20 views

Backwards finite difference for mixed partials at higher order

I'm trying to understand what is the general method for calculating a backwards difference for a mixed partial of $n$ variables. Let's start with one variable: The forward and backward finite ...
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24 views

Ideas on how to improve stability in solving PDE

I am solving a non-linear second order system of PDEs in two variables. The equations are too complicated to write out here, but an essential feature is that there is a propagating wave which then ...
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10 views

Stability of Difference Equations

Suppose $\dot{x}(t) = h(x(t))$ is a well posed ODE and $x^*$ denotes an asymptotically stable equilibrium point. Let $B$ be an open set containing $x^*.$ Also suppose that $B$ lies in the region of ...
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18 views

Proper writing of IBVP PDE & Finite Difference Implementation

I've seen a few examples (see slide 4 ) of the 2D heat equation described as $$ \begin{cases} u_t(t,x,y) = \nabla^2 u(t,x,y), \quad t > 0, \quad (x,y) \in (0,L) \times (0,H), \\ u(t,0,y) = ...
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78 views

What is this type of function called? How can I translate it to a different origin?

A factory produces 1 widget per week. A builder builds 1 factory each week. A construction firm trains 1 new builder each week. Partially-produced things do not produce anything. Starting with 1 firm, ...
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1answer
25 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
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36 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...
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6 views

What is the parameter of the Streamline-Diffusion Method?

Recently, I want to apply streamline-diffusion(SD) method to solve Burgers Equation. \begin{align} u_t + uu_x - \varepsilon u_{xx} &= 0,\\ u(0,t) = u(X,t) &= 0,\\ u(x,0) &= u_0(x),\ \ \ ...
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42 views

Expressing a function's value using finite differences

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and let $x = (x_0, x_1, x_2, \dots)$ be a sequence of pairwise distinct real numbers. For every $n \in \{1, 2, \dots\}$ and every ordered $(n+1)$-tuple ...
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25 views

How to make a 9-point two-dimensional stencil for a elliptic operator?

I want use a finite difference schem to discretizate the elliptic operator: $$ \nabla \cdot \left( k(x, y) \nabla p\right), $$ where $k(x, y)$ a positive scalar function and $p$ is the unknow. We can ...
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26 views

Stability of a scheme for one way wave equation.

Multiplying the scheme \begin{equation*} \dfrac{v_m^{n+1} - v_m^n}{k} + \dfrac{a}{2}\left(\dfrac{v_{m+1}^{n+1} - v_m^{n+1}}{h} + \dfrac{v_m^n - v_{m-1}^n}{h}\right) = 0 \end{equation*} by $v_m^{n+1} + ...
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24 views

Show that $\Delta^{-1}\sin\alpha x = - \cos(\alpha x - \frac{α}{2})/(2\sin \frac{\alpha}{2}) + c(x)$ for fixed $\alpha\in \Bbb{R}$.

Show that $\Delta^{-1}\sin\alpha x = - \cos(\alpha x - \frac{α}{2})/\left(2\sin \frac{\alpha}{2}\right) + c(x)$ for fixed $\alpha \in \Bbb{R}$. I can find a few tables for anti-difference tables ...
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49 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
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27 views

Laplace equation in polar coordinates [closed]

Hy, I'm new in here but i have a question please help me. I want to solve a Laplace PDE in a polar coordinate system with finite difference method, but I have a problem with boundary conditions at r = ...
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38 views

Given this operator what is inverse operator?

Given operator $$\Delta_{sym}[f(x)]=\frac{f(x+\varepsilon)-f(x-\varepsilon)}{2\varepsilon}$$ what is inverse operator in terms of summations? For instance, given operator ...
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527 views

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 ...
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28 views

How to transform $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ into a system of first order PDE's and finite difference matrix

So I have this equation: $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^3 u}{\partial x^3}$ and I need to transform it into a system of first order PDE's. I was thinking like this: ...
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1answer
25 views

In the finite difference formulas, how can we pick h to give a certain tolerance?

There's lots of questions on here about finite differences. In particular, picking the 'best' h value. But what if I want to find the biggest 'h' which bounds to a given tolerance? On first glance, I ...
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14 views

Finite Differences Scheme for Poisson Nernst Planck Equations

I want to solve numerically a simple version of Poisson Nernst Planck equations in 1-D on the domain $[0,\ell]$: $C_t=-j^c_x=(C_x +\rho\phi_x)_x$ $\rho_t=-j^\rho_x=(\rho_x +C\phi_x)_x$ ...
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28 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
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1answer
21 views

Proof that strictly tri-diagonally dominant matrix has an inverse

We are given the following theorem of which we need only know the result. Theorem Suppose an $n\times n$ matrix $A= (a_{ij})$ is tri-diagonal with $a_{i,i-1}a_{i,i+1} \neq 0$, for each ...
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32 views

Finite Difference method for nested derivatives

I'm moderately experienced with finite difference methods, but I'm hoping that somebody has better intuition than I do. Suppose I have the following expression which I want to evaluate via finite ...
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1answer
52 views

Construct a symmetric Poisson matrix from $\nabla \cdot (\rho\nabla p)$

Let $\rho$ and $p$ be 2D scalar fields. How do I construct a symmetric Matrix $A$ so that is satisfies \begin{equation} Ap = \nabla \cdot (\rho\nabla p) \end{equation} in finite differences? I am ...
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22 views

Computing the CFL number for non-linear advection

Consider $u_t -(1-2t)u_x=\phi(t,x) $ with $u(x,0)=u_0(x)$ for integrable $\phi$. The solution for this PDE is given by $ u(x,t)= u_0(x+t+t^2) + \int_0^t \phi(x(z),z) dx$. For a consistent scheme ...
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1answer
20 views

Writing the implicit scheme using matrices

\A= \begin{bmatrix} 2 & -1 & 0 & . & . & . & 0 \\ -1 & 2 & -1 & . & . & . & . \\ 0 & . & . & . & . & . &. \\ ...
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62 views

Closed form solution to $\frac{1}{a-1}= \log a$

I want to find a function that satisfies $$\Delta [f(x)]=f'[x]$$ Obviously the solution is the exponential function $f(x)=a^x$ with $a$ in between $2$ and $e$ because $\Delta[2^x]=2^x$ and ...
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1answer
86 views

Infinite sum involving ascending powers

I was working on a problem involving finite differences and came across the following sum as a general solution to a problem I was working with $$ 1 + x + \frac{1}{1 + a}x^2 + \frac{1}{1 + a} ...
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18 views

Finite differences

I don't know finite differences but I have reached a step in a proof where I need them. In particular, I need to know the $3k$th differences of $4\cdot 1^{3k}, 4\cdot 2^{3k},...,4\cdot (3k)^{3k}$. I ...
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81 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) ...
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50 views

Solution to the heat equation using a finite difference scheme

I have used a difference scheme and Fourier Analysis to find an expression for the solution to the heat Equation. My problem is plotting my solution. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ ...
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55 views

Laplace problem with elliptic BC - FINITE DIFFERENCES

I am experiencing some trouble in solving the problem $u-\Delta u = f$ with periodic boundary conditions. First question: Is the problem always solvable? If I am not mistaken $-\Delta u = f$ ...
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26 views

Order of accuracy of an scheme

How does one find order of accuracy for the following scheme (see picture below). The picture shows two equations instead of the typical one equation which confuses me a lot. You can read about the ...
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38 views

Irregular grid for finite differences PDE solution

I got a project in a class I'm taking. In this project, I need to solve fluid dynamics and heat equations. Up to here, the problem is not so complicated, however the chamber shape is the problem: as ...
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1answer
46 views

Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$ f'(n) = \frac{f(n+1) - f(n)}{1} ? $$ Such a discrete ...
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12 views

Discretization of Variable Coefficients

I'm trying to discretize something of the form $$ a(x)\frac{df}{dx}. $$ Now, with $x_{i}\equiv i \Delta x$ and $a_{i}\equiv a(x_{i})$ (similarly for f), this is what I think it should be: $$ ...
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74 views

Approximate Laplace Operator with Central Difference in Polar Coordinates

I'm trying to approximate the Laplace operator in polar coordinates with the central difference quotient and I know how to do this in cartesian coordinates, but with polar coordinates I just feel ...