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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6
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1answer
206 views

Does mean removal increase accuracy of numerical differentiation?

I wish to compute the derivative of a vector through numerical differentiation. Let's say, we use a standard 2nd order central difference scheme, to arrive at a differentiation matrix, and apply it on ...
5
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1answer
1k views

FEM: which is the correct way to impose Dirichlet B.C

I know the Neumann B.C. is implicit in FEM language. However, I have seen at least two ways to impose Dirichlet B.C. e.g. for the following problem 1D, $$\nabla^2 u + \nabla u= 0, u_{left}= 1, u_{...
5
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1answer
262 views

Convergence of Jacobi's method for a semilinear elliptic PDE

I have an iterative finite difference scheme for the Poisson equation $$ \nabla^2 u=-\rho $$ It's the Jacobi method, which has the form (for 1D systems) $$ u^{n+1}_{i} = \frac{1}{2}(u^n_{i+1} + u^n_{...
5
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1answer
2k views

Stability of numerical method for 1D Burger's equation

I am trying to solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also ...
4
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1answer
269 views

Points on the interface

We consider the problem $\left\{\begin{matrix} k(x)\Delta u(x)=f(x) & \text{ in } \Omega\\ u=0 & \text{ in } \Gamma \end{matrix}\right.$ where $\Omega \subset \mathbb{R}^2$ open and ...
4
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1answer
2k views

Numerical solution of non-linear diffusion equation via finite-difference with the Crank-Nicolson method

I want to numerically solve the non-linear diffusion equation: $$ \frac{\partial}{\partial t} T(x,t)= \frac{\partial}{\partial x}\left(T^{5/2} \frac{\partial T}{\partial x} \right) $$ I want to use ...
4
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2answers
3k views

Testing 1D Poisson Solver

I'm trying to test a simple 1D Poisson solver to show that a finite difference method converges with $\mathcal{O}(h^2)$ and that using a deferred correction for the input function yields a convergence ...
4
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2answers
1k views

choice of the norm for the error of the numerical method

When I read books on finite differences they often end up using discrete $L^2$ norm for estimating the error as it naturally arises from weak formulation. I was wondering if people do that in Sobolev ...
3
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1answer
650 views

Finite difference method basic implementation on Octave

Trying to study the error of FDM for a second order derivative versus step size I calculated the coefficients and validated them, but the output has errors for small step sizes. The function in ...
3
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1answer
1k 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: $(T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0$ ...
3
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3answers
468 views

Numerical approximation for a known exact solution of advection-dispersion equation

My goal is to create a numerical solution of 1D-solute transport (Convective-dispersion equation, CDE) to match it's analytical solution based on experimental data. The CDE can be written as (where, C=...
3
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2answers
378 views

Finite Difference and Finite Volume as special cases of Finite Element

I have heard this many times that "FD is a special case of FE" and separately, that "FV is a special case of FE". However I have never come across a brief document that substantiates that claim, ...
3
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1answer
153 views

Stable method for solving a HJB equation

I am considering finite difference methods and their error analysis for solving HJB equation of the following form: $$ v_t=|\sigma(x)v_x|,\quad x\in \mathbb{R}, $$ where $\sigma$ is a given function ...
2
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1answer
1k views

Derivatives Approximation on non uniform grid

I was trying to approximate 1st derivative of a function $\phi$ on a non uniform grids: basically my aim is to do this on a uniform grid on the same domain, so I can calculate it on the "new" grid. ...
2
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1answer
359 views

Analytical Solution of Transport Equation

I'm looking at the analytical solution of the convection-diffusion equation $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ with initial ...
2
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0answers
71 views

Efects from the boundary in advection equation [duplicate]

I am implementing the advection equation $u_x+(1/c)u_t=0$ following a Crank-Nicholson finite difference scheme. The equation for this is \begin{eqnarray*} -\frac{\gamma}{4} w_{n-3 j+1} + w_{n-2 j+1} ...
1
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1answer
142 views

FEM oscillations for polynomials of degree 1

I have the following eliptic 1-D problem $$-\mu u'' + \beta u' = 1$$ $$u(0) = u'(1) = 1$$ where $\mu = 10e^{-5}$ and $\beta = 1$. For this specific problem I am using the following space steps $h=[0.1,...
1
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1answer
107 views

Crank-Nicholson for diffusion-advection vs diffusion equation

Let's consider the following 1D diffusion equation: $\frac{\partial u}{\partial t} = xk \frac{\partial}{\partial x}(\frac{1}{x}\frac{\partial u}{\partial x})$ where we assume that the diffusion ...
1
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1answer
543 views

Crank-Nicolson method for inhomogeneous advection equation

Suppose we have the inhomogeneous advection equation $$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$ for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
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2answers
324 views

Mass conservation in 1d diffusion by method of lines

I am solving the 1D diffusion equation by discretization using the method of lines. My problem is that I don't manage to ensure mass conservation. I have read many similar questions about the topic ...
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0answers
75 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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0answers
1k views

Crank-Nicolson for 2nd- and 4th-order finite differences

I modeled the heat equation, $$ u_t = au_{xx} $$ using the common 2nd-order Crank-Nicolson scheme, $$ \frac{u^{n+1}_i-u^{n}_i}{dt} = \frac{a}{2\,dx}\left(u_{i-1}^{n+1}+u_{i+1}^{n+1}-2u_i^{n+1} + u_{i-...
1
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1answer
318 views

Solving first versus second order PDE

I am trying to numerically solve a PDE, and just had a question as to the validity of a certain approach. For example, given the PDE: $$ \frac {\partial ^2 E}{\partial t^2} = - k\frac {\partial E}{\...
0
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1answer
1k 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 ...
0
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1answer
102 views

Question on comparing the accuracy of numerical schemes

This is a follow up to my previous post here I'm solving the following 1D transport equation . $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$...
0
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1answer
5k views

Python Finite Difference Schemes for 1D Heat Equation: How to express for loop using numpy expression

Hello all, I've recently been introduced to Python and Numpy, and am still a beginner in applying it for numerical methods. I've been performing simple 1D diffusion computations. I suppose my ...
0
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1answer
157 views

How to compute matrix representation of $\hat{y}\frac{\partial}{\partial x}$?

I have a 2-dimensional system which I would like to solve numerically (I'm using finite difference method right now), and its an eigenvalue problem. I have a term that looks like $H\psi(x,y) = [-\frac{...
0
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1answer
129 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 ...

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