# 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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### 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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### 2nd order centered finite-difference approximation of $u_{xy}$

The problem is to find a 2nd order finite difference approximation of the partial derivative uxy, where u is a function of x and y. Page 5 of this pdf I found does a centered difference approximation ...
646 views

### Conservative finite-difference expression for the advection equation

Following on from the earlier question I am trying to derive a finite-difference scheme for the advection equation which is conservative. It was suggested that for advection equation with variable ...
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### CFL condition in polar coordinates

In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads $$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$ ...
226 views

### How do I analyze the error for the Crank-Nicolson method on a parabolic PDE?

I would like to do the analysis for the Crank-Nicolson method on a non-uniform grid for the parabolic equation with variable coefficients. I was able to prove everything for a uniform grid by energy ...
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### 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 ...
4k views

### Poisson equation with Neumann boundary conditions

I'm trying to solve the Poisson equation with pure Neumann boundary conditions, $$\nabla^2\phi = \rho \quad in \quad \Omega\\ \mathbf{\nabla}\phi \cdot \mathbf{n} = 0 \quad on \quad \partial \Omega$$...
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### Trouble implementing Neumann boundary conditions because the ghost points cannot be eliminated

Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question. I ...
207 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 ...
7k 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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### method of frozen coefficients and its relation to von Neumann stability analysis

I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show ...
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Can anyone point me to methods for varying $h$ in gradient estimation for noisy numerical optimization? Some programs have the user give a fixed $h$, which is used for forward-difference or central-...
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### eigenvalue analysis vs fourier analysis for stability and their equivalence

I have a question regarding stability analysis for constant coefficient pde. Suppose I am looking at the pde $$u_t= au_{xx}$$ So the first approach is to compute the amplification factor, and this is ...
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### Numerical Solution to Schrödinger Equation--Multiple Wells

I am trying to solve for the allowed wavefunctions and energies for a 1D quartic potential well. To do this I am using the patching method (https://engineering.dartmouth.edu/microeng/otherweb/...
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### 9-point stencil finite difference Laplacian with variable diffusion coefficients

So I'm trying to implement a 9-point stencil discretization to the 2D difussion equation. The stencil is here. However, most of the literature deals with a Laplacian that has a constant diffusion ...
3k 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 ...
294 views

### How to design good finite difference schemes? [closed]

In principle, I know finite differences. In university, we discussed it and derived consistency and boundary conditions. But I am still left with a big question. How to design a good finite ...
3k views

### Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
629 views

### Interpolation schemes to move data between cells and nodes

I work on non-graded quadtree grids where the entire grid is a hierarchy of cells specified using a quadtree data structure, where, in general, there is no constraint regarding the relative size of ...
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### What are the negatives of using higher order finite diference schemes?

I was looking at this wikipedia page: http://en.wikipedia.org/wiki/Finite_difference_coefficient It is a lists of higher order finite difference approximations, is there any negatives in using these ...
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### PetSc vs Sundials for serial numerical computations?

I am currently working on a physics problem that turns into a non-linear boundary value problem. I need an efficient numerical solver that I could run on my laptop with i5 dual core CPU. I am ...
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### 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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### I'm having trouble debugging multigrid. What to do?

I've spent far too much time coding and debugging multigrid. While I clearly can't post all of my code as it would be silly to ask someone to go through all that code, is there anything I should pay ...
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### Finite-difference discretization for a convective term

How does one discretize the classical convective term in a transport equation using finite differences? I know the finite volume schemes out ther i.e. upwind, central differencing etc. Are there ...
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### 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 ...