# 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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### Linearized implicit time stepping

Consider the general FD implicit time stepping scheme $\frac{x_{t+1} - x_t}{\Delta t} = f(x_{t+1})$, where $x$ is the vector variable of interest and $f$ is some function, generally non-linear. ...
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### Finite difference scheme for "wave equation", method of characteristics

Consider the following problem $$W_{uv} = F$$ where the forcing term can depend on $u,v$ (see Edit 1 below for the formulation), and $W$ and its first derivatives. This is a 1+1 dimensional wave ...
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### Implementing a finite difference method in Mathematica

I am trying to iterate the following equation $$x_{k}(n+1)=x_k (n)-\epsilon (x_{k+1}(n)-2x_k(n) +x_{k-1}(n))+\sqrt{\epsilon}\; \eta_{k}(n)$$ where $n$ denotes which time step I'm on and $k$ is the ...
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### Boundary conditions for the advection equation discretized by a finite difference method

I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs. The books and notes which I currently have access to all say ...
651 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 ...
2k views

### Finite difference coordinate transformation for spherical polar coordinates

I have part of a problem that is described by the momentum conservation equation: $\frac{\partial \rho}{\partial t} + \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}(\rho u \sin \theta) =0$ ...
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### Smoothing the diffusion coefficient to improve convergence

I have been reading a book by Thomee and he considers the case of $u_t=(au_x)_x$, for the case of $a$ possibly being discontinuous. Then he says that the problems with convergence might occur, and ...
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### analyze stability on a nonuniform grid

Assume you have a stability constraint between the space distance in time and space, for example, with an explicit Euler method for $u_t=u_{xx}$ we know $\tau\leq h^2/2$. That is, one can do stability ...
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### how to measure the error of a finite difference method

Suppose I am solving a pde with a solution known with a finite-difference method. I can represent it as $A_hu_h=f$ for some approximating matrix $A_h$. And I define the discrete norm in which I will ...
566 views

### 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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### How to choose a stable PML for pseudo-spectral method with strongly varying velocity

My friend was working on this, and he asked me about the stability of PML while applying on pseudo-spectral method, I believe his concern was how to choose the difference(if the difference should be ...
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### regularity of a solution and its affect on the global error

I am solving the following equation $u_t=x^2u_{xx}+\frac{x-y}{T-t}u_y$ with an initial data $u_0=max(y-C,0)$ for some $C$ in the domain of the numerical method. In time I am solving that on $[0,T]$. ...
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### 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 ...
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### Impact of irregularity at the boundary on the error analysis

I am looking at the pde of the type $u_t=\mathcal{L}u$ for some elliptic operator $\mathcal{L}$ and on some domain $D$. Assume I am solving that with a finite difference method and want to estimate ...
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### 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 ...
829 views

### Why is my second order accurate method only converging at first order when the coefficients are rough?

I have a method that is supposed to be second order accurate based on asymptotic analysis/Taylor series expansion which assumes that the solution is smooth. I am solving a PDE which has very rough ...
266 views

### Defining electric current source excitations for surface integral equation formulations

In a finite difference (FD) based electromagnetic formulation based on a Yee cell grid, one can define electric current source excitations ($J$) on the $E$ field grid points. At a distance, the fields ...
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### I don't understand how to correctly calculate truncation error

I am looking at the finite difference methods to solve simple $u_t=a(x,t)u_{xx}$. There are explicit, implicit, Crank Nicolson. The latter is said to be more accurate since the local truncation ...
771 views

### How can I derive a bound on the spurious oscillations in the numerical solution of the 1D advection equation?

Suppose I had the following periodic 1D advection problem: $\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$ $u(0,t)=u(1,t)$ $u(x,0)=g(x)$ where $g(x)$ has a ...
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### How do you improve the accuracy of a finite difference method for finding the eigensystem of a singular linear ODE

I am attempting to solve an equation of the type: $\left( -\tfrac{\partial^2}{\partial x^2} - f\left(x\right) \right) \psi(x) = \lambda \psi(x)$ Where $f(x)$ has a simple pole at $0$, for the ...
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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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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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### How can I obtain a one dimensional finite difference formula for $U_{xx}$ with unevenly spaced nodes?

I know that if I had evenly spaced points, I can use $U_{xx}\approx \frac{U_{i-1}-2U_{i}+U_{i+1}}{dx^2}$. But if my gridpoints are unevenly spaced, I assume that I can obtain the finite difference ...
831 views

### Largest eigenvalue of FD discrete Laplacian

Is there good approximation for largest (in magnitude) eigenvalue for discrete Laplacian ($\nabla^2$) obtained from nonuniform structured grid (like that)? Of course, one can always use general ...