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# 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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### Conservative formulation for compact finite difference schemes

At the Section 4.2 of this paper (which is very well known in the computational fluid dynamic community), the author claims that it is enough, for the compact finite difference formulation in eq. 4.2....
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### Discretisation of logarithmic derivative: Deriving the formula

I'm reading a paper where they use a discrete approximation of a logarithmic mass growth rate as follows: $$\frac{d \log M}{d \log t} \approx \frac{(t_B + t_A)(M_B - M_A)}{(t_B - t_A)(M_B + M_A)}$$ ...
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### Limit to volume change in a discretized mathematical model?

I have set up a mathematical model describing the diffusion of ozone out of a gas bubble. The bubble is surrounded by a thin gas film. So actually, the model describes the diffusion of ozone through ...
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### How to account for the interface between two different phases in a discretized diffusion model?

I have tried to set up a model for the diffusion of a gas into a liquid. The two media are next to each other and the geometry is spherical because the system should simulate the diffusion out of a ...
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### 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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### 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 ...
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### Why naively chopped finite difference matrix works for different ODE boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
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### Why do I still obtain a unique solution with one-sided formula when b.c. isn't enough?

Let me illustrate the issue with an simplified example. Suppose we want to solve the following problem with finite difference method (FDM): \frac{\partial u(t,x)}{\partial t}=\frac{\partial^2 u(t,x)...
We have a 2D electromagnetic field (in the sense that: $E=(E_x,E_y,0)$, $B=(0,0,B_z)$, and all derivatives with respect to $z$ are $0$), and we are considering a system made up of two walls at $x=-b$ ...
Assume that we have a 2D electromagnetic field (in the sense that: $E=(E_x,E_y,0)$, $B=(0,0,B_z)$, and all derivatives with respect to $z$ are $0$), and that we are considering a system made up of two ...