# 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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### Should the derivative of an array be calculated array by array or element by element in CFD codes?

I am making my own finite difference computational magnetohydrodynamic code in Fortran 90. Looking at other codes they appear to calculate for example their $x$-derivatives, bb of their variables, e.g....
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### numerical instabilities in Fluid Dynamics, Finite Element Method

I'm looking for references to understand where the numerical instabilities come from in hydrodynamics in general, and notably when the Péclet number: $Pe>1$. I'm using the finite element method. ...
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### Predictor-Corrector vs. Deferred Difference Corrections

I want to use the Numerov method but keep higher-order terms from the Taylor expansions. In the literature, I found the term "Deferred Difference Corrections" for the procedure of first solving the ...
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### Why do many people use FDM method to solve Stokes equations, i.e., saddle point matrix?

For numerical methods of the Stokes equations, with appropriate boundary: $$-\nabla^{2} \vec{u}+\nabla p=\overrightarrow{0}$$ $$\nabla \cdot \vec{u}=0$$ one may use FDM (finite difference method) ...
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### Implementing structured grid boundary conditions using NumPy arrays?

I am making a toy code in Python to solve the advection equation $$u_t + cu_x = 0$$ with, for example, periodic boundary conditions. Background information The numerical grid is specified like this: ...
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### Implementation of boundary conditions for 1D Euler equations

I'm trying to solve 1D Euler equations with gravity in spherical coordinates using a finite-difference TVD MacCormack method on a non-uniform grid of $N$ components, following the method provided in ...
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### WENO scheme on curvilinear coordinates

I've been developing a curvilinear FVM code. So far I've implemented the PPM scheme and am looking into adding WENO schemes. So far I've been discretizing the grid metrics using a second-order central....
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### Need an example Legendre-Gauss-Radau pseudospectral differentiation matrix or Matlab code

I'm trying to implement various kinds of pseudospectral methods for direct optimization in Matlab using IPOPT. I've got some working Legendre-Gauss-Lobatto code, but would like to use the flipped ...
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### Dirichlet boundary conditions in the 1D Heat Equation

Please consider the assignment I have uploaded on the picture. I am confused about the functions $g_L(t)$, $g_R(t)$ and $\eta(x)$. What are they and how do I find them... My question: Is it possbile ...
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### High-accuracy numerical differentiation

I have a $200 \times 200$ matrix representing the values taken by a function over an equally spaced grid. I would like to perform derivatives on it. I am interested in its gradient (i.e. its ...
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### Error for the finite differences scheme — Advection equation

Consider the advection equation (1D in space) $$\frac{\partial u}{\partial t} + V\, \frac{\partial u}{\partial x}=0$$ and we solve it numerically on $[0,1]\times [0,1]\ni (t,x)$ using a forward ...
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### Approximation Error in a Finite Difference Approximation of the Square of Derivative

First Part: (First-order derivative) Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is \begin{align}\tag{1} f(x + h) = f(x) + hf'(x) + \frac{1}...
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### Oscillations when solving parabolic heat equation with FTCS

I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...
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### Numerical solution to N-dimensional diffusion on simplex?

Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
Consider a Boundary Value Problem: $$\delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b$$ $\delta,a,b$ are known parameters. I want to implement Newton'...
I would like to solve the following differential equation numerically in 2D, $$\frac{\partial z^-}{\partial t}+(\vec{B}\cdot\vec{\nabla})z^-=0,$$ see Wikipedia if you are curious about what the ...