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

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Collocated Grid Navier Stokes Solver

I want to solve Navier Stokes equations on a collocated grid. Earlier, I was using a MacCormick scheme based solver where I discretized predictor step in forward differences and corrector step in ...
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1answer
51 views

How to set the temperature at the vertices points for a rectangular domain?

Suppose I have to solve the 2-D heat equation in a rectangular domain using the finite difference method, for the boundary conditions say: $T_1$ is the temperature of the right side of the rectangle, ...
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1answer
117 views

Three body problem in C++

I am in a begginers programming course and we got a little project. I chose to simulate the three body problem using the Euler method. Even though the system is chaotic there are some special cases ...
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1answer
90 views

Second order interpolation scheme

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
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49 views

Gauss-Seidel iteration weighted by change

In general, I can use Gauss-Seidel iteration for finite difference solution of partial differential equations. In my case I am only solving an analog of steady-state heat transfer, so there is no ...
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1answer
87 views

How to define residual in multigrid approach?

I wish to solve the two-dimensional Navier Stokes equations using multigrid method on a collocated grid using the Predictor-Corrector method mentioned below. But first, let me elaborate on what I had ...
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1answer
114 views

Discretize Poisson equation with derivative of delta function as source

Consider the PDE \begin{equation} \frac{d^2}{dx^2} g(x) = \frac{d}{dx} \delta(x-x_0), \end{equation} with $x, x_0 \in [0,1]$ and $g(0)=g(1)=0$. What is the best method to discretise the derivative of ...
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110 views

Unwanted Oscillation in FDM simulation of elastic wave equation

I am using staggered grid FDTD for solving elastic wave equation. A description of which can be found at (geodynamics.usc.edu/~becker/teaching/557/reading/Virieux1987.pdf). I have generated a ...
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1answer
122 views

Is this finite difference approach correct?

I am solving incompressible 2D Navier-Stokes equations with zero y-component velocity. The geometry is a simple 2D pipe of a length $L$ and diameter $W$ and there is only two boundary conditions: ...
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1answer
98 views

Stability Criterion for this Explicit Scheme

I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as: $$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial ...
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69 views

CFD implementation in software [closed]

I've been learning CFD theory for 4 months (which includes FEA, FDM AND FVM) and now I want to start running simulations. As a novice to CFD software, I would appreciate some advice on where to start, ...
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133 views

how to solve a 2D non-linear Poisson equation?

I am trying to solve the following equation for $P(x,y)$: $$I = P \nabla \cdot \frac{1}{P^2} \nabla{P}$$ where $P$ and $I$ are functions of x and y. $I(x,y)$ and $P(x,y)$ I want to understand ...
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1answer
23 views

Suitable method for simulation of in-fiber interferometer

I am trying to simulate an optic-fiber sensor (in-fiber interferometer) to study its respond to temperature. The method I am using is finite-difference time-domain (FDTD), and I come out with a large ...
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0answers
62 views

Solving Lid-driven Cavity on a Collocated grid

I want to solve the two dimensional lid driven cavity flow on a collocated grid. I already have used a staggered grid and validated my results with Ghia (1982). Recently, I came across this paper ...
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1answer
59 views

Boundary conditions generalized eigenvalue problem

Consider the following eigenvalue problem \begin{equation} \mathcal {L} x(s) = \lambda x(s), \end{equation} where \begin{equation} \mathcal {L} = \alpha \partial^4_s + (s^2-1)\partial^2_s + s ...
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1answer
120 views

How do I program periodic boundary conditions? [duplicate]

Hi I have a code below that solves non linear coupled PDE's given Dirichlet boundary conditions. However I need to implement periodic boundary conditions. The periodic boundary conditions are ...
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34 views

Representing a 3D system in 2D (Electromagnetic modelling)

Ok so I'm a complete beginner in computational modelling (I use analytical methods of physics typically) but I would like to model an anisotropic, aperiodic (but not random) finite array of metallic ...
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1answer
77 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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161 views

What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I am reading a paper [1] where they solve the following non-linear equation \begin{equation} u_t + u_x + uu_x - u_{xxt} = 0 \end{equation} using finite difference methods. They also analyse the ...
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1answer
57 views

Boltzmann equation and equilibrium distribution function

The free-streaming term in stationary 1D Boltzmann equation satisfies the equilibrium distribution function, that is: \begin{equation} \text{Fr}\{f\} := v(k)\dfrac{\partial f(x,k)}{\partial x} - ...
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1answer
69 views

Finite difference scheme for Webster equation

Webster equation is a popular generalization of 1D wave equation used for ducts of variable cross-section $S \equiv S(x)$. Assuming harmonicity in time, the spatial equation for propagation of ...
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74 views

The numerical solution of a (very ugly) set of integro-diferential equations

I've been having fun playing around with a simple model for reaction in a co-flow fluidized bed (I'm afraid I'm a chemical engineer, not a computer scientist). Unfortunately, simple physical ...
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124 views

Finite difference recursion and higher order

This may be a trivial question, but I've always wondered... For the classical, central finite difference schemes, if I'm interested in determining the second derivative, does applying the first ...
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39 views

Using Periodic BC for Taylor Green Vortex Flow Simulation

I want to solve the very simple case of two dimensional Taylor Green Vortex Flow. I am using incompressible Navier Stokes equations and Artificial Compressibility Method as my solver where I introduce ...
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48 views

Periodic boundary conditions for solving Navier Stokes Equations on a Staggered Grid

I want to solve two dimensional Navier Stokes equations on a staggered grid for the case of Taylor-Green Vortex. My initial conditions are standard sine and cosine functions. As I am aware, I should ...
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122 views

Heat equation from implicit scheme with Neumann B.C

To solve the heat equation from implicit scheme subject to Neumann boundary condition we can write: $$ T_i^{j+1}-T_i^{j}=\alpha (T_{i+1}^{j+1}-2T_{i}^{j+1}+T_{i-1}^{j+1}) $$ $$ \textbf{A} T^{n+1} = ...
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1answer
85 views

Stability in discretization of a PDE

Suppose I want to numerically solve for $f(x,k)$ the one-dimensional Boltzmann equation for electrons in steady-state condition, given as: \begin{equation} \left( \dfrac{\hbar k}{m} \right) ...
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3answers
231 views

Numerical simulation of a reaction-diffusion system on MATLAB with finite difference discretization of spatial derivative

I have a model of a system which consists of diffusing reactants and intermediates. For each variable, $u_i$, the final representative equation looks like the standard reaction-diffusion form: ...
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51 views

PDE discretization (via finite difference sheme) question

So after posting this question and reading all your comments I would like to make this new question (update). If you consider the three equations presented here: $$\frac{\partial \rho}{\partial t} ...
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1answer
55 views

Non-uniform finite difference Adaptive Mesh Refinement

Assuming that the crosses in the figure below are unknowns in a vertex-centered finite difference scheme in an Adaptive Mesh, how can I calculate the double derivate (Laplacian) at the Red x ? The Red ...
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71 views

discrete definitions of curl $\nabla \times F$?

I have some data defined in an array (an image) and I need to find the curl of a certain function. Wikipedia has an integral definition of curl that I like, maybe it can be discrete. $$ \nabla ...
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83 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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1answer
115 views

Implementing Finite Difference Adaptive Mesh Refinement code

For a start I need to implement a 2-D, vertex centered, finite difference scheme, Adaptive mesh refinement serial code. I have the following doubts before starting: Is the input to AMR (say 2-D) ...
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1answer
97 views

Generating a non-uniform grid

I am interested in generating a 1D non-uniform grid on the interval [0, L] with N points, where a region of width $\sigma$ and ...
3
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2answers
143 views

Are the Finite Difference and Finite Volume Methods different after the application of the Gauss Divergence theorem on the FVM?

I have a rudimentary question about the differences between finite difference (FDM) and finite volume methods (FVM). In FDM we concentrate on the nodes (points) in space while in FVM we concentrate ...
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1answer
126 views

1D uniform flow test case for compressible flow

Are there any one dimensional hyperbolic test cases for a 1D uniform flow? (I need the test case for the purpose of showing the satisfaction of geometric conservation law on a moving mesh).
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1answer
71 views

Implementation of Neumann boundary condition with method of lines - 1D diffusion/reaction equation

I am solving the monodimensional diffusion/reaction equation by discretization using the well-known method of lines ${\partial c\over\partial t}=D{\partial^2 c\over\partial x^2}+ r\text{ for ...
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46 views

Solving a nonlinear problem with CDF

I'm trying to solve this problem: $\begin{cases} \partial_t E=-k\left([f(\rho)-i.\left[\delta+\frac{1}{2}a\left[\dfrac{\nabla^2_{\bot}}{4}+1-\rho^2\right]\right]]E - 2CP\right)\\ \partial_t ...
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1answer
163 views

Poisson equation finite-difference with pure Neumann boundary conditions

I'm trying to solve a 1D Poisson equation with pure Neumann boundary conditions. I've found many discussions of this problem, e.g. 1) Poisson equation with Neumann boundary conditions 2) Writing the ...
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1answer
107 views

How to calculate dispersion relation from a Finite Difference (FD) wave simulation

I have a python code that calculates the solution of the inhomogeneous acoustic wave equation for a 2D medium with any velocity and source configuration. It was implemented using Finite Differences ...
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1answer
132 views

Finite-difference approximation of the 2nd derivative operator matrix for a staggered grid

I'm working on a computational physics assignment and I was looking for some help as I've got stuck! The question is: Write a function to create the finite-difference approximation of the 2nd ...
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177 views

Solving the transient heat convection diffusion equation (2D) using BTCS in Matlab with derivative BCs

I'm really stacking guys on how to solve the Solving the transient heat convection diffusion equation (2D) using the full implicit BTCS in Matlab with derivative Boundary condition and variable ...
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17 views

Converting events with irregular arrival times into a continous signal (1st, 2nd and 3rd order)

I got events coming in at irregular intervals. From these I need to emit the number of events arrived and the expected first and second derivative over some specified measurement period P. So ...
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1answer
117 views

How efficient (compared to “normal” methods) is using a sparse finite difference matrix to solve differential equations?

Any differential operator can be written as a finite difference matrix acting on a vector of values. I recently used it to solve $\Delta A = j$ by writing the Laplacian as a finite difference ...
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1answer
141 views

Heat Equation - PDE

I'm trying to model the Black-Scholes Equation (transformed into a heat equation) using method of lines in Python. The transformed formula is basically \begin{equation*} \frac{\partial u}{\partial ...
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138 views

CFL Condition and Convection Diffusion Equation in 2D

I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of ...
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76 views

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 the normals of some surface that has been non uniformly sampled. How would ...
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102 views

Are FEM or DGFEM methods based on integrals or PDEs?

I know that FVM is based on the integral form of conservation laws, and FDS is based on PDEs. What I'm confused by, is whether FEM and DGFEM formulations are based on integral or pde form of ...
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2answers
101 views

Analytical Solution to the acoustic / scalar (Inhomogeneous) wave equation with source term

The acoustic wave equation in 2D is $$\frac{\partial^2}{\partial t^2}p(x,z,t) = c(x,z)^2\left[\frac{\partial^2}{\partial x^2}p(x,z,t) + \frac{\partial^2}{\partial z^2}p(x,z,t)\right] + s(x,z,t) ...
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1answer
109 views

Calculation of error

I have written a code in which I find the approximation of the solution of this elliptic problem. I calculated the error using the following part of code: http://pastebin.com/7b5mmuRW but I get the ...