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I did the 1st example in Schneider's online book both ways and got the same graphs, so I'm convinced the answer is that they are equivalent but that Yee's algorithm is faster by a factor of 4 (it uses 1/2 as many points and 1/2 as many time steps).

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You don't have to discretize your problem (XY model). For each step, just take some value as the new $\theta$, and calculate the transition rate accordingly. Of course, when choosing the new value of $\theta$, better don't do it in a completely random way, otherwise your transition rate might be usually too small and you are just wasting time. Having said ...

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A few off the top of my head: Double pendulum is a nice example of chaotic motion and quite simple to model. Ising model of a magnet can provide a good introduction to modelling quasi-random processes. Solving the heat equation with various combinations of sinks/sources. Lots you could do with optics from very simple to very complex. Also lots you could do ...

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Like mentioned in the comments, you got the initial velocity wrong. You source says: Further to this, the selected separation strategy foresees a fixed separation velocity of approximately 0.187 m/s. This imposes limitations on the capability of directing Philae towards the comet, i.e. it restricts the domain of possible positions and velocities ...

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Your equation in the Liouville form is elementary for numerical integration, it is structurally just a linear advection equation with spatially varying coefficients. The transformed equation with the kernel F is not useful at all for numerical solution, don't bother with it. All we have here is a 2D advection equation (I use y instead of p): $\partial_{t} ... 3 You are asking how to numerically solve a second order initial value problem. An initial value problem involves advancing some initial state over time given an ordinary differential equation (ODE) that describes the time evolution of the state. There are many books, journal articles, and college classes about this topic. There is no one perfect technique. ... -1 The light ray in general relativity travels along the null geodesic, which is determined by the simple equation $$g_{\mu \nu} (x) \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = 0,$$ where$g_{\mu \nu}$in your case is the Schwarzschild metric. Using this equation and the initial condition (angle$\alpha$) you should be able to trace your light ray back ... 2 As said in this answer, velocity and position are not varied independently. Indeed, when deriving Euler-Lagrange equations, we explicitly use the fact that$\delta v=\frac d{dt}\delta x$. So, when I add the constraint$v_i=\frac{x_{i+1}-x_i}{\Delta t}$, specifying$x_1$and$x_n\$ remains the only additional thing to converge to the solution. For example, ...

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