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

learn more… | top users | synonyms

0
votes
0answers
47 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 ...
1
vote
0answers
32 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} - ...
0
votes
0answers
49 views

How one can explain to have non-unique numerical solution at the interface boundary?

If the existed computational domain is divided into two pieces and developed numerical scheme is applied to these lets say right and left side domains of the interface individually to obtain two ...
1
vote
1answer
61 views

A doubt in Multigrid V-cycle

Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0. Relax $u$ on $Au = b$ at level 2 for 3 times. Find residual $r2$ at level 2, then restrict to ...
1
vote
2answers
113 views

How can I prove that two eigenvectors are orthogonal?

I obtained 6 eigenpairs of a matrix using eigs of Matlab. How can I demonstrate that these eigenvectors are orthogonal to each other? I am almost sure that I ...
-2
votes
1answer
46 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
0
votes
1answer
78 views

Finite Difference method

I'm following the article at http://www.paykin.info/irina/project_2.jsp Finite Difference method. How to interpret this one? How to convert this to pseudo code? $$u(i,j+1) = 2u(i,j) + ...
7
votes
1answer
65 views

Stability analysis of Heun's method

I am using Heun's method with a third order upwind spatial scheme, which is suggested by Shao (2008) to be used for solving the horizontal advection part of the advection-diffusion equation. This is ...
2
votes
0answers
35 views

Rank deficient Jacobian in discretized periodic solutions to autonomous ODE

I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting ...
1
vote
1answer
79 views

openfoam - Programming customized PDEs

I am looking for a method to automatically solve custom PDEs on a custom control volume. Specifically I would like to solve equations similar (but not exactly alike) to: $$\frac{\partial y}{\partial ...
1
vote
1answer
73 views

Correct way of computing norm $L_2$ for a finite difference scheme

I am computing the rate of convergence of my finite difference scheme in norm $L_2$. Which is the correct way to compute it? This: \begin{align} L_2 &= ...
2
votes
1answer
88 views

Why a finite difference scheme would give second order of accuracy in norm L2 but 1.5 with L1 (while 1 with Linf)?

My finite difference scheme for the 2D Euler equations is second order accurate in theory, since all the terms are second order accurate, with the advective terms being third order. So I expect a rate ...
0
votes
0answers
53 views

The centered difference operator for fractional function

Recently, I come to a question about the 2nd order centered finite difference approximation of a fractional function, more precisely, we set $\delta^{2}_x u(x,t) = u(x_{i-1},t)-2u(x_i,t) + ...
2
votes
2answers
61 views

Integro-differential PDE and method of lines

Let us consider the following equation : $$\partial_t u = \Delta_x u + K*u$$ where $K$ is a smooth kernel, $u(x,t)$ is the unknown and $x$ is in $\Omega$ a domain in 1d or 2d. I want to numerically ...
0
votes
0answers
40 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 ...
1
vote
0answers
90 views

Tips on improving stability in numerical scheme for non-linear 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 ...
6
votes
3answers
96 views

Stability criterion for waves in anisotropic solids

The equations of motion for an elastic solid are given by $$\begin{align} &\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \rho \ddot{\mathbf{u}}\\ &\boldsymbol{\sigma} = ...
0
votes
1answer
79 views

Spectral Coefficients of Implicit Finite Difference Solution

This is something I've been trying to figure out for a long time, and all I have is vague numerical results. I'm trying to answer the following question analytically: Suppose I have a time dependent ...
2
votes
2answers
159 views

The meaning of conservative discretization in Galerkin FEM and Discontinuous Galerkin

I do understand the meanning of "conservative discretization" within the FVM/FDM framework, indeed it is well explained in this post. Now, according to the table in this slide (pp.4), it concludes: ...
4
votes
1answer
81 views

Need a good reference for numerical transport phenomena

I'm a chemical engineering undergraduate and I'm currently starting to work in a theoretical transport phenomena/colloid science group. While my group has a nice code base for larger scale ...
3
votes
0answers
59 views

2D wave equation with Mur boundary condition - setting up the RHS and solving (time-steps)

I am trying to solve a 2D wave equation implicitly using FD with central approximations with the following boundary conditions $$\begin{align} &u=2\sin\left(\frac{2\pi}{5}t\right)\quad \text{at ...
0
votes
1answer
118 views

Solving PDE with state and time dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
0
votes
2answers
40 views

How to obtain streamslines from velocity field data points

I managed to solved a lid-driven cavity flow using LB code. It gave me the velocity field data points. Now I have to obtain streamlines too, of course from the obtained velocity field. Besides, I ...
2
votes
0answers
101 views

Boundary equations for constant right hand side in Poisson equation (FD)

I am setting up to solve a 2D Poisson equation $\nabla^2u = T$ by the finite difference method. I am using the multigrid method. On the coarsest mesh, where the grid size is 2x2, I want to set up ...
2
votes
1answer
152 views

Can Gauss-Seidel/SOR (preconditioned?) be applied to a non-diagonally dominant matrix?

After applying finite difference method to a Laplace/Poisson problem always arises a diagonal dominant system of equations that can be solved with Gauss-Seidel or SOR methods. If the original PDE does ...
3
votes
0answers
75 views

Computing accuracy of my finite difference scheme for uniform grid on a non-uniform grid

I have a ADI finite difference scheme for the 2D Navier-Stokes equations that uses a second order accurate (central) approximation for the advective terms. I am ignoring the diffusive terms for now. I ...
1
vote
3answers
285 views

2D Stokes equation Code

Does anyone know where could I find a code (in Matlab or Mathematica, for example) for he Stokes equation in 2D? It has been solved numerically by so many people and referenced in so many paper that I ...
0
votes
0answers
27 views

Book recommendation on simulation of linear and nonlinear electronic circuits

I am looking for a book on electronic circuit simulation explaining the numerical modeling using transient and frequency based methods from a computational point of view. It should also have an ...
2
votes
0answers
97 views

1 D Diffusion equation FDM with different layers

I'm trying to solve this particular equation $\frac{\partial u}{\partial t} = \frac{\partial}{\partial x} \big[D_{i}(x)\frac{\partial u}{\partial x} \big] + S(x,t)$ where the $i$ index denotes ...
4
votes
2answers
145 views

Usability of upwind finite difference schemes

NOTE: I asked this on Mathematics Stack Exchange and there were no answers. So, I thought I might try here. Upwind schemes like the classic "upstream" scheme, can be used to solve, for example, the ...
0
votes
0answers
12 views

Is there a commonly accepted measure for SNR degradation from a derivative operator?

Is there a commonly accepted measure for the impact of a derivative operator (compact FD or otherwise) on signal or image SNR? I am about to do this empirically, taking SNR measurements on a large ...
4
votes
0answers
89 views

numerical analysis of a partial integro-differential equation

I have to numerically solve a nonlinear partial integro-differential equation. This is my equation, $$\frac{\partial y(x,t)}{\partial t}=\int_{-\infty}^\infty K_0(|x-u|) \frac{\partial^2 ...
0
votes
0answers
46 views

Help about Fluid-Fluid coupling techniques

I need a little help and advice with a project I want to do: the idea is to "couple" (I don't know whether I can call it like this) a conservative Navier-Stokes Solver (Fractional-step, 2nd order FDM) ...
1
vote
0answers
69 views

Visualization of solution for a MAC grid

So I implemented the projection method as in Chorin (1969) for the solution of the Navier-Stokes Equations with the Boussinesq Approximation, using a 2nd Order accurate FD scheme. The primitive ...
1
vote
0answers
130 views

What is the difference between SSPRK3 and RK3 time discretization methods?

Here, SSPRK3 refers to third order strong stability preserving Runge-Kutta and RK3 refres to regular third order Runge-Kutta method. The meaning of the method is obvious from the name. However there ...
4
votes
1answer
88 views

How to evaluate a series of derivatives?

Consider the function $$f(\mathbf{x}) = \sum_{n=0}^{N} a_n \left( (\mathbf{b}-\mathbf{x})\cdot \nabla \right)^n \frac{1}{r}$$ where $r = |\mathbf{x}| = \sqrt{(x-x_0)^2 + (y-y_0)^2}$ and $a_n$ and ...
0
votes
0answers
35 views

multivariable ode

I have a system of pde which I dicretized along the spatial derivative to yield a time-dependent ode. The resulting ode is multivariable,highly non-linear and stiff.Also they have largely different ...
2
votes
1answer
142 views

Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
5
votes
2answers
130 views

How to impose boundary conditions on eigenfunction problems?

I am trying to solve for the eigenfunctions of a (1D) differential operator using finite differences: $$A \, f(x) = \lambda f(x)$$ Here is an example in Python where $A = \partial_x^4$: ...
2
votes
1answer
55 views

Modified diffusion equation and unstabilities

I am trying to simulate the phase separation of a binary mixture. If the free energy F is known as a function of the concentration $c$, the dynamical equation is: $ \frac{\partial c(x,t)}{\partial ...
1
vote
0answers
147 views

eigs routine in octave

I am using octave and observed a problem with the eigs-routine for non symmetric matrices. Using GNU octave version 3.8.1 the code below gives significant difference of eigenvalues although same ...
0
votes
0answers
65 views

How to compute convergence rate for FVM in matlab?

I have implemented a finite volume scheme using Lax-Friedrichs flux and using Roe's linearization. I want to compute the order of convergence vs. the space-step $h$. I have computed a solution with a ...
0
votes
0answers
146 views

CFL condition and Lax-Friedrich numerical flux

I got confused when trying to implement a scheme using Lax-Friedrichs numerical flux for a system of equations in 1D. According to my notes Lax-Friedrichs numerical flux is $$f_{LF}(u_l,u_r) = ...
1
vote
1answer
61 views

methods for a peculiar BVP system

Consider the following system defined on the open interval (-1, 1): $y_1' = c y_3 \\ y_2' = c y_4 \\ y_3' = -f(y_1, y_2)y_2 \\ y_4' = f(y_1, y_2)y_1 $ given $ y_3(-1) = 0 = y_3(1) \\ ...
1
vote
1answer
58 views

Solving first-order ODE with Dirichlet b.c. using CDS -> singular matrix

I decided to solve using finite differences one of the simplest differential equations which has an analytic solution, notably $\frac{du(x)}{dx}=2x$. The equation is first-order ODE with quadratic ...
2
votes
0answers
42 views

2nd order differences for modelling a physically discrete phenomenon?

Consider the diffusion of some unknown $R$ on a physically discrete, 1-D ring: x---------x---------x---------x n = 0 1 2 3 We let ...
0
votes
1answer
119 views

Implement Robin Boundary Condition

This is a follow up to this question. I have a nonlinear BVP on $x=0$ to $L$: $$ (T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0 $$ to which I apply a ...
3
votes
2answers
256 views

Neumann Boundary Condition at r=0 in Polar Coordinates (Numerical BCs)

I have asked a question in this regard earlier. I am trying to solve the following equation in Polar Co-ordinates: $$ u_t - (u_{rr} + \frac{1}{r} * u_r + \frac{1}{\theta} * u_{\theta\theta} + bu) = ...
0
votes
0answers
91 views

Finite Difference in Polar Co-ordinates

Background Please note that I am duplicating the question on scicomp. I have already asked this in math. I am trying to come up with a scheme in Polar Co-ordinates for the following PDE: PDE I am ...
2
votes
1answer
232 views

Solving a Nonlinear BVP using Finite Difference Method

I am trying to write a code to solve a nonlinear BVP using the Finite Difference Method. The BVP is: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...