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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2answers
584 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
113 views

Trouble Implementing 1d Wave Equation Finite Difference Solver

Im trying to solve the 1d Wave Equation on $x \in \mathbb{R}, t > 0$: $$u_{tt} = c^2u_{xx}, \hspace{5mm} u(x,0) = \cos(4 \pi x), \hspace{5mm} u_t(x,0) = 0$$ with $c = 1$ and a periodic boundary ...
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2answers
1k views

Is the maximum/minimum principle of the heat equation maintained by the Crank-Nicolson discretization?

I'm using the Crank-Nicolson finite difference scheme to solve a 1D heat equation. I'm wondering if the maximum/minimum principle of the heat equation (i.e. that the maximum/minimum occurs at the ...
7
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4answers
2k views

Finite Difference Method Stability

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
6
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1answer
208 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
2k 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
308 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
3k 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 ...
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1answer
826 views

Thomas algorithm for 3D finite difference

For 1D finite difference, the resulting linear system is tri-diagonal and can be solved in O(n) using the Thomas algorithm. I am trying to solve a finite ...
4
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1answer
276 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 ...
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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 ...
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3answers
524 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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1answer
183 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 ...
3
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2answers
445 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, ...
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1answer
2k 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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1answer
899 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 ...
2
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1answer
118 views

Confusion about Zabusky and Kruskal's stepper for the KdV equation

In Zabusky and Kruskal's paper about solitons, they derive the following update for the Korteweg de Vries equation (their footnote 6): \begin{align*} u_{i}^{j+1} = u_{i}^{j-1} - \frac{1}{3} \frac{k}{...
2
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1answer
108 views

Is this a valid way to implement Neumann BCs in finite differences?

I'm trying to solve the 1D heat equation with Neumann boundry conditions numerically using finite differences: $$u_t = \alpha u_{xx}$$ $$u_{x}(0, t) = u_{x}(L, t) = k$$ $$u(x, 0) = u_0(x)$$ The main ...
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
468 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
75 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} ...
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1answer
676 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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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-...
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0answers
78 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
146 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
144 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,...
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2answers
428 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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2answers
350 views

Implementing routine for $-\nabla\cdot (k(x,y) \nabla u)=f$ in Matlab

I am solving the Poisson Equation for 2D given by the following expression: $$-\nabla\cdot (k(x,y) \nabla u)=f$$ in a rectangle with Dirichlet conditions on the boundary using Matlab. In principle I ...
0
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1answer
172 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{...
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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
135 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 ...
0
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1answer
142 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
8k 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 ...

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