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

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69 views

Can Gauss-Seidel/SOR (preconditioned?) be apllied to all zero diagonal system?

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
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0answers
47 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
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3answers
179 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 ...
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0answers
25 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 ...
1
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0answers
40 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
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2answers
119 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 ...
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0answers
11 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 ...
3
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0answers
82 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 ...
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0answers
41 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) ...
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0answers
38 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 ...
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0answers
109 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
85 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 ...
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0answers
29 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 ...
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0answers
21 views

Am I using cyclic reduction to parallelize this algorithm?

I am attempting to implement a modified anisotropic-diffusion filter on the GPU. The methodology I describe here including the equations and code listing are taken from this paper . The author of ...
2
votes
1answer
122 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 ? ...
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2answers
111 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
48 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 ...
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0answers
100 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 ...
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0answers
44 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 ...
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0answers
74 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) = ...
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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) \\ ...
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1answer
53 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 ...
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41 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 ...
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1answer
95 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
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2answers
155 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) = ...
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74 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
170 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: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...
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0answers
44 views

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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1answer
107 views

Solve steady state reaction-diffusion/Helmholtz equation numerically

I am solving a problem of the form: $\dfrac{\partial u(x,y,t)}{\partial t} = \nabla^2 u(x,y,t) - f(x,y,t)u(x,y,t) - \kappa(x,y,t)$ At the moment, I am solving this at each time step by assuming a ...
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0answers
44 views

Finding a scalar field in order to generate a solenoidal vector from a given vector

I am working on generating a (complex) solenoidal vector field $\mathbf{A}$ from a prescribed (complex) vector $\mathbf{a}$ and the gradient of a scalar, $b$, such that $$\mathbf{A} = \mathbf{a} + ...
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1answer
88 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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1answer
40 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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3answers
121 views

Solve diffusion equation with linear source term

I would like to solve numerically the diffusion equation, where the sink term depends linearly on the field, and there is field-independent sink: $\frac{\partial^2 u(x)}{\partial x^2} =f(x)u(x) - ...
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2answers
92 views

Numerical Solution of non-linear diffusion equation using Finite Differencing

I'm trying to solve the following non-linear diffusion equation: $$ \frac{\partial}{\partial t} u(x,t)= \frac{\partial^{2}}{\partial x^{2}}u(x,t)^{3}$$ $$ -1\leq x \leq1, t \geq 0 $$ with the boundary ...
4
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2answers
169 views

Finite Difference Error Estimate for an Elliptic PDE with an Oscillatory Coefficient

Suppose I want to solve the PDE: $$-\nabla\cdot\left( a_\epsilon\left(x\right)\nabla u\right)=0 \text{ in } \Omega$$ $$u=g(x) \text{ on }\partial\Omega$$ Here, I assume that ...
0
votes
1answer
86 views

Neumann boundary problem

I'm writing a solver for a differential equation with two neumann boundaries (u'(0)=u'(1)=0) and I can't figure out how to determine how to solve the problem. What will my boundaries be and how do I ...
2
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0answers
103 views

Finite Volume/difference scheme for 2d continuity equation

I am trying to solve the following 2D continuity equation on the rectangular domain [0,1]x[0,1] using the finite volume method. $\rho_t + \nabla \cdot (\rho v) = 0$ where the velocity $v =(v1,v2)$ is ...
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votes
1answer
131 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; ...
4
votes
2answers
290 views

How can I prove numerical diffusion in upwind scheme for transport equation

I was just implementing the upwind scheme for a linear transport equation $u_t + cu_x = 0$ where $c=0.5$ and I saw that the solution was indeed advected but over time it starts to diffuse. Can anyone ...
2
votes
2answers
74 views

Infinite Function Value on Dirichlet Boundary

I have been working on a multigrid solution to a non-homogeneous Dirichlet boundary value problem. However, the function goes to infinity on the boundary. This causes numerical overflow errors to be ...
3
votes
2answers
76 views

How does constraint resolution affect the stability/accuracy of numerical integration?

I understand some basic analysis techniques (local truncation error, global error, zero-stable, absolute stable, etc.) of numerical integration. But I find it hard to apply these techniques in ...
4
votes
2answers
392 views

Crank-Nicolson method for solving nonlinear parabolic PDEs

Is the Crank-Nicolson method appropriate for solving a system of nonlinear parabolic PDEs like $\partial u/\partial t - a\Delta u + u^4 = 0$ ? I tried to apply this method for solving such system but ...
2
votes
1answer
314 views

Finite differences scheme for 2D advection equation

I'm trying to study with a finite difference method the 2D advection equation with a space-dependant flow. Taking a function $f(x,y,t)$ solution of the equation : $$ ...
4
votes
1answer
83 views

Error implementing Robin boundary conditions in toy ODE problem

I am attempting to solve the following ODE problem: $$-u''+ u = x$$ $$u(0) = 0$$ $$u'(1) = -u(1)$$ The exact solution is: $u(x) = e^{-x-1} - e^{x-1} + x$ I have a Dirichlet at $x = 0$ and a Robin ...
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1answer
118 views

Get symmetric Finite Difference matrix in non Laplacian settings

I would like to solve a system of differential equations $u+\nabla(\nabla\cdot u)=f$ or in more detail $a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$ ...
2
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0answers
92 views

2D Neumann Conditions on Irregular Domain

I would like to model the 2D diffusion equation with Neumann BC's inside the following egg-shaped domain: I would like to use the finite difference method with the discretization implied by the ...
3
votes
1answer
189 views

Numerically solve a PDE in Python with a term calculated by coarse-graining

I'm trying to solve a PDE in Python of the form, $\dfrac{\partial c(\mathbf{x}, t)}{\partial t} = \mathrm{D} \nabla^2 c(\mathbf{x}, t) -\gamma \rho(\mathbf{x}, t) c(\mathbf{x}, t)$ where $c$ ...
0
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1answer
132 views

Solve a fourth order differential equation

I want to solve $$ \frac{\partial^2}{\partial t^2}u(z,t) + a\frac{\partial^2}{\partial z^2}u(z,t) + k\frac{\partial^4}{\partial z^4}u(z,t) = 0 $$ with $u(z,0) = 1+0.1e^{-\frac{z^2}{2}}$. I'd like to ...
2
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1answer
162 views

Solve a differential equation with finite difference method

I want to solve this equation $$ -\frac{1}{2}f''(x)+2a\ f(x)^3 = f(x)\mu $$ One exact solution (there are a lot of different kinds) of this equation is $f(x) = f_\infty \tanh(\sqrt{2a}f_\infty x) $ ...
2
votes
3answers
111 views

Computing the derivative on a mesh

I have a 2-D mesh of triangles and I have a scalar function $f(x,y)$ defined at all the vertices of this mesh. I want to accurately estimate $\frac{\partial f}{\partial x}$ and $\frac{\partial ...