# Questions tagged [finite-differences]

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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### The approximation of the first derivative at x in order 3

I need to approximate the derivative $u'$ at the point $x$ by taking the points $x−h$, $x+h$ and $x+2h$ into account, and the order should be 3. we suppose that the function $u$ is regular. My ...
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### Discretization of $\nabla\cdot (u\nabla v)$ for chemotaxis

I am considering the following set of PDEs: \begin{align} \frac{\partial u}{\partial t} &= f(u,v) - \chi \nabla\cdot \left(u\nabla v\right)+d\Delta u\\ &= f(u,v) - \chi\nabla u\cdot \nabla v -...
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### Numerical analysis for PDE using finite difference method

How we can solved 2nd part to obtain unique solution for the above scheme i start firstly integrating both sides to get $u'(a)-u'(b)=\int_{a}^{b} f(x)dx=0$ Now how i will continue ??
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### orthogonal eigenvectors of Toeplitz matrix.

Let $A\in\mathbf{R}^{NxN}$ be a matrix of the form \begin{pmatrix} 2 & -1 & 0 & \cdots & \cdots & \cdots & \cdots & 0\\ -1 & 2 & -1 & 0 & & & &...
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### Developing a Finite Difference Scheme for a non-linear PDE

I came across this question while studying for an exam: Consider the equation $$u_t = u u_{xx} + uu_{x}$$ to be solved for $t > 0$, $0 < x < 1$, with $u(0,t) = u(1,t)$, $u_x(0,t) = u_x(1,t)$...
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### Eigenvalues of Laplace operator under Neumann boundary conditions

I'm trying to numerically calculate the eigenvalues of the Laplace operator under Neumann boundary conditions utilising a 5 point stencil approximation scheme. Starting with the case of Dirichlet ...
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### Solve heat transport equation numerically with forward finite differences and explicit timestep

I am working on numerical solutions to the diffusion equation and came across a counter-intuitive phenomenon. Let's stay in 1D for this. The diffusion / heat transport equation is ($f$ my state ...
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Consider the following heat equation: $$u_t=u_{xx} \quad\text{ on }\quad(0,\pi)$$ with mix boundary condition $u'(0)+u(0)=0$ and $u'(\pi)+u(\pi)=0$. The eigenfunctions of the laplacian are $$e^{-x}... 0answers 9 views ### Discretization of differential of a map on discrete manifolds. Suppose N and M are two regular submanifolds of \mathbb{R}^3. Suppose you discretize these as meshes, and for simplicity assume they have same number of vertices and connectivity. Suppose F:N\... 0answers 14 views ### Finite difference method applied to non-linear equation I'm trying to understand my professor's notes regarding applying the finite difference method to non-linear equations. We're told that, we're given the equation u_{t} = u_{xx} + f(u), where f is a ... 0answers 26 views ### Coefficients for the three-dimensional discrete Laplacian I am trying to numerically solve u_{xx} + u_{yy} + u_{zz} = 0 using a 27-point stencil, meaning each node in my finite difference domain interacts with every node in a 3 \times 3 \times 3 cube ... 0answers 23 views ### Why is the residual vector after one Gauss-Seidel sweep on a red-black ordering equal to 0 in the black-corresponding nodes? Working with finite difference methods here. One possible ordering of the gridpoints is red-black. This results in a system matrix of the form \begin{pmatrix}A_{RR} & A_{RB} \\ A_{BR} & A_{... 0answers 38 views ### Stabilizing Upwinding of Transport Equation with Varying Velocity and Large Gradients I am attempting to solve the following transport equation using a 2D finite-difference scheme$$c_t+\mathbf{v}(c)\cdot\nabla c=\frac{1}{Pe}\nabla^2 c,$$where c is modeling a local concentration, ... 1answer 179 views ### Uncertain how to set up the Neumann boundary conditions for Heat Equation Implicit FDM To smooth stock price data, I am currently looking to apply the heat equation$$ u_\tau - \kappa u_{xx}=0 $$with Neumann boundary conditions$$ \frac{d^2u}{dx^2}(x_{min}+\Delta x, \tau)=0 \\ \frac{d^...
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Thank you in advance for any help with this assignment. I promise, I have spent hours trying various sources to understand these terms and operators, but I just cannot make the fundamental leap ...
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### Numerical Scheme for Heat flow between two materials with different conductive properties.

I am trying to write a simulation in MATLAB for a $1D$ heat flow problem in which there are two materials in contact with each other but with different thermal properties. The results are unstable so ...
I want to compute Dupire's local volatility, but I'm struggling since several days. Here is the formula to get the local variance, with $y=\ln \left(\frac{ K}{F} \right)$ and $w=\sigma_{BS}^2\,T$, and ...