# Questions tagged [numerics]

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### Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
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### About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation

I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$: $$A u + B^t p = f$$ $$Bu = g$$ where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
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### Is there a graphical interpretation or explanation of automatic differentiation compared to numerical differentiation

I have been looking at automatic differentiation for solving differential equations lately. I understand the basic ideas of using Dual numbers and such for finding derivatives, etc. However, I feel ...
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### What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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### Ritz projection error estimate for bilinear Lagrange element?

For second order elliptic problem \begin{align} -\Delta u=f,\quad in ~~\Omega\\ u=0,\quad on~~ \partial\Omega, \end{align} we have for the Ritz projection for $P_1$ conforming element \begin{align} \|...
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### Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?

I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
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### How to implement point source or volume source in finite element implementations

I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe. I have points positioned along the length of the pipe (blue dots in the image above). I ...
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### Another way to evaluate the gravitational force from a uniform cube?

Appendix A of Liu, Baoyin, and Ma (2011) Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube shows an analytic expression ...
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### How important is learning hardware/architecture for scientific computing?

Apologies if this is a bit of a soft, unclear, or opinion-based question. I'm a relatively new PhD student in a (computational) quantum chemistry group. My group develops and maintains a few software ...
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### Numerical algorithm to calculate stream of discounted cash flows

Is it possible to calculate the flow of cash in each period given a certain fixed net present value with a numerical algorithm? I would like to be able to apply a certain weighting that can be applied ...
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### Interpreting multivariable root-finding results from Matlab's fsolve algorithm

Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
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### For a hyperbolic PDE, is there any proof that the BDF2 method is stable for integrating them?

I would like to ask a question on the stability of BDF2 applied to hyperbolic PDEs. Say I have a hyperbolic equation as $\frac{\partial c}{\partial t} + {\bf U} \cdot {\bf \nabla}c=0$. This system is ...