# 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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### why is complex step differentation better than central differences

I came across this post at Cleve's Corner, where he shows that complex step differentiation is more accurate than central differences. The error in both methods is $O(h^2)$. So why exactly does ...
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### Stability condition FCTS method

The FTCS method comes from the discretization of a diffusion PDE like this: $$a^{2} \frac{u_{i+1}^{k}-2 u_{i}^{k}+u_{i-1}^{k}}{\Delta x^{2}}=\frac{u_{i}^{k+1}-u_{i}^{k}}{\Delta t}$$ If I have the ...
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### Howo to implement complex step derivative for complex functions?

I have a complex analytic function of which I want to take the numerical derivative. \begin{align} f(z) &\equiv f(x,y) = u(x,y) + i v(x,y) \\ \frac{d f(z)}{d z} & = \lim_{h \to 0} \frac{f(z + ...
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### Stability of the Forward-Time Central Space method, section 9.3 in LeVeque

I am reading section 9.3 in Leveque about stability. The explanations are very short and brief so I am asking for some help to understand and elaborations. The result of the section is that for FTCS ...
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### How to discretize a non-linear PDE with boundary conditions and intial value

Consider this non linear PDE: $$u_t + c(u^2)_x =\alpha u_{xx} \text{ , } -1<x<1 , t>0$$ with $$u(-1,t) = g_L(t) \text{ , } u(1,t) = g_R(t) \text{ and } u(x,0)=f(x)$$ where the 3 functions(...
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### Comparison between stability and accuracy of various Finite Difference schemes

Im Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy. I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
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### Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered

Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-...
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### discretizing advection equation with variable wave speed + stability

I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form ...