# Questions tagged [finite-differences]

A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.

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### Express eigen values of Gauss Seidel matrix?

In matrix form the Gauss Seidel iterations can be expressed as: $$X^{m+1} = (D + L)^{-1}\textbf{b} - (D + L)^{-1}UX^{m}$$ Where $A = D + L +U$ is a finite difference matrix expressing the original ...
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### What is three-point backward difference?

I fully understand two-point, forward/backward/central difference. I know them mathematically and graphically but I'm not quite sure what the three-point backward difference is. I know that we get the ...
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### Compact Finite Differences for the Heat Equation with Robin Boundary Conditions

I am trying to solve the Heat equation with Robin Boundary condition: $$u_t(x,t) = u_{xx}(x,t), \\ u(x,0) = g(x), \\ u(0,t) + u_x(0,t) = h_0(t), \\ u(1,t) + u_x(1,t) = h_1(t)$$ for $0\leq x\leq1$ ...
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### Predictor-Corrector Method (FDM, FVM)

I was searching to find an example predictor-corrector solving method applied either to FVM or FDM scheme regarding either fluid or heat flow. Do you have any ideas where I can find some examples? ...
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### Von Neumann stability for inhomogeneous PDE

I've got an inhomogeneous PDE of the following form: $$\alpha\partial^2_xu+\partial_tu=f$$ with $\alpha<0$ and a source term $f$. I descretise $u$ according to $u_{m,n}=u(m\Delta t,n\Delta x)$ ($f$ ...
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### Application of Crank-Nicholson method to higher order time derivatives

In literature, the Crank-Nicholson method is always used to solve only the heat equation. The heat equation contains a first order time derivative. I never read in any literature that the Crank-...
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### What is a sufficient condition for stable convergence of a variable coefficient partial differential equation solution solved by the FDM?

While studying the finite difference method for solving partial differential equations, I came across this chapter of a book. On the first paragraph of page $2.27$ it says that the Von Neumann ...
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### Literature request on the subject of finite difference method for solving PDEs with variable coefficients

I am trying to understand how to solve PDEs with variable coefficients using the finite difference method. I read a few books about numerical solutions to PDEs, but none discuss how to solve a PDE ...
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### PDE numerical stability condition can not be determined by using the Von Neumann stability analysis

The Von Neumann stability analysis is used to determine the stability conditions of finite difference schemes. However, the functions that are contained in the PDE are restricted to only two ...
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### Order of a numerical iteration method

Suppose that we have the definition of order $p$ of a numerical method as in the first snippet below. Now I want to prove that for a one-point iterative method this order $p$ is a positive integer. To ...
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### Finite difference method solves the wave equation for one set of bounday conditions, but does not when I change them

I wrote a finite difference algorithm in Matlab to solve the wave equation which is derived here. When I ran my code, the plotted graphs of the numerical and analytical solution deviated, which is the ...
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I'm a little bit confused about how to understand Butcher tableaus for implicit methods, when I have a matrix. Say I have some ODE that is defined by $$\frac{\partial \mathbf{y}}{\partial t} = \... • 53 0 votes 0 answers 41 views ### Convert difference equation to differential equation Suppose I have the following third-order difference equation: Y[t]+aY[t-1]+bY[t-2]+cY[t-3]=G d^t I need to convert it into a differential equation and I followed this method: y' = y(t+1)-y(t); y'' = y'... 1 vote 0 answers 13 views ### Under which conditions does (v-w)>(x-y) imply (v*a-w*b)>(x*c-y*d)? I know that (v-w)>(x-y) (v-w), (x-y) > 0  (a-b)<(c-d) (a-b), (c-d)<0 v,w,x,y,a,b,c,d>0 Can I conclude from this that (v*a-w*b)>(x*c-y*d)? • 41 0 votes 1 answer 15 views ### Finite Difference discretization of squared transient term If we were to discretize with Finite Difference the following transient term:$$\frac{\partial u^2}{\partial t},$$would it be okay to write$$\frac{(u^{j+1})^2-(u^j)^2}{\Delta t}. Or does this not ...
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Suppose I have an elevated Delaunay triangulation such as below: Suppose the vertices are embedded as $(x_i,y_i,z_i) \in \mathbb{R}^3$ where $i \in \{1, \cdots, m \}$. Let us assume that $z(x,y)$ is ...