Questions tagged [finite-difference]

Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.

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Finite differences for incompressible viscous fluid equations

I am working with the equations for incompressible viscous fluid: $$ \partial_t \vec{\omega} + (\vec{u}\cdot\nabla)\vec{\omega} = \nu\nabla^2\vec{\omega} $$ $$ \nabla^2 \vec{\psi} = -\vec{\omega} $$ $...
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319 views

3D Diffusion Equation in Fourier space

I'm solving the 3D Diffusion equation $$u_t=k(u_{xx}+u_{yy}+u_{zz})$$ in MATLAB using Fourier techniques. I assume a 3D Fourier expansion $(e^{-ipx},e^{-imy},e^{-imz})$of the solution. Physical ...
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76 views

Nonlinear 2D modeling of Neural Electromagnetic field in Matlab

I am trying to replicate the MATLAB simulation presented in this paper. More specifically, I have to code the solution to this equation $$\frac{a}{2R_i}\frac{\partial^2 V_m}{\partial x^2} - \left[g_\...
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62 views

Quick scheme for separable first-order ODE

I'm trying to integrate an incredibly simple ODE: $$ y'(x) = -f(y),\quad y(0) = y_0 \ , $$ from $x=0$ to $x=1$. This is a decay type of equation, $f$ is the (variable) decay rate and $y$ is the ...
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What is a reasonable manufactured solution to test finite difference method? [duplicate]

What is a reasonable manufactured solution to test the following equation against its finite difference approximation? I want it to look like a Cosine function about $0$, rotated about $Z$ axis (...
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111 views

Computing multiple numerical derivatives at once

Lets say I have a function $f(X) = f(x_1,...,x_N)$ to be integrated. But unlike time discrete methods, my integrator uses quantisation to advance time, that is if $|x - q| > dQ$, with $q$ being the ...
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Computation of potential flow using dirichlet conditions [duplicate]

I know I already posted this question and I thank you for your answers, but unfortunately I didn't find what I was looking for among them. Anyway now I'll rewrite the question more clearly, because ...
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48 views

Howo to implement complex step derivative for complex functions?

I have a complex analytic function of which I want to take the numerical derivative. \begin{align} f(z) &\equiv f(x,y) = u(x,y) + i v(x,y) \\ \frac{d f(z)}{d z} & = \lim_{h \to 0} \frac{f(z + ...
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434 views

Finite Difference for Fourth-Order PDE

How to discretize the following 4th order PDE using finite difference method? $$\frac{\partial^{2} y}{\partial t^{2}}+\frac{\partial^{4} y}{\partial x^{4}}=0$$ thanks
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645 views

Stability analysis for coupled nonlinear system of partial differential equations

I'm trying to solve a nonlinear partial differential equation \begin{equation} L(u_{xxtt},u_{xx}u_{tt},u_{xt}^2,u_{xt},u_{tt})=0 \end{equation} using finite difference methods. In order to remove the ...
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385 views

How to initiate spirals in this model?

I am trying to reproduce Tang & Othmer paper which is related to excitations and oscillations in G-protein model in Dictyostelium discoideum, an amoeba species. The mathematical model in the paper ...
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147 views

Can you give some information for rothe method [closed]

I want to learn a numerical method for PDEs other than finite difference method. After some research on internet i have found Rothe method and it looks interesting to me. Unfortunately, i couldn't ...
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Explicit scheme for heat equation with Neumann boundary conditions in Maple

$\displaystyle \frac{\partial u}{\partial t}=\alpha(x,t)\cdot \frac{\partial^2 u}{\partial x^2}+b(x,t)$ $u(x,0)=f(x)$ Initial condition $u_x(0,t)=0$ 2nd type Boundary condition $u_x(1,t)=0$ 2nd ...
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72 views

Numerical solution of the advection equation with Crank–Nicolson finite difference method

I need to implement a numerical scheme for the solution of the one-dimensional advection equation $$\\\frac{\partial u}{\partial t} + C(x, t) \frac{\partial u}{\partial x} = 0 \\\\$$ $$ \\ C(x,t) = \...
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98 views

Finding derivative of Matrix at different grid points using Finite difference methods/ Cholesky Factorization

I want to code this problem in MATLAB. It would be a huge help if someone can suggest to me how I can approach it. I need to solve the below-highlighted equation, I need ...
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843 views

Any note on Immersed boundary finite difference method?

For parts of a talk, I need a note on "Immersed boundary finite difference method", mainly about the reason of appearing this branch in the finite difference methods, considering mathematical ...
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1k views

explain the difference between 1D Poisson solvers

I compared 2 methods for 1 dimensional Poisson equation solution. One is finite-difference method, "Successive Overrelaxation" from http://www.cs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html; ...
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308 views

Adding Non-Linear source term to 2d Implicit MATLAB code

I'm running out of time for this code so any help would be greatly appreciated. I am currently coding the 2D heat/diffusion equation in matlab but i'm having trouble adding in the source term. my ...
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73 views

How to avoid negative concentration from numerical solution using FDM scheme?

$\frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} + w \frac{\partial C}{\partial x} = D \left(\frac{\partial^2C}{\partial x^2}+\frac{\partial^2C}{\partial y^2}\right)-C \cdot \left(\...
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201 views

Use Finite Difference Discretization to find approximate solution to the Poisson's equation

I've just been introduced to the Poisson's equation. I've never had the need to dealt with PDE, so I'm a bit lost. Apparently we can compute an approximate solution of the Poisson's equation $$\frac{\...
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Successive over-relaxation formation of heat equation?

What is the form of SOR iterative equation for the heat equation $u_{xx}=u_{t}-1$ using centered differences both in time and spatial derivatives and using Crank-Nicolson method? $$(u(x,0)=u(L,t)=u(0,...

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