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Our partial differential equation (PDE) and boundary conditions (BCs) are: $$T_t=\alpha T_{xx};$$ $$T(0,t)=30\text{ and }T_x(L,t)=0.$$ Let's use a generic initial condition (IC): $$T(x,0)=f(x).$$ First, we transform the dependent variable $T(x,t)$: $$u(x,t)=T(x,t)-30.$$ This means that: $$\Rightarrow u(0,t)=30-30=0.$$ The derivative $T_x(L,t)$ isn't affected,...


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Yes. Symplectic integrators are designed to do exactly that. They are intended for solving Hamiltonian systems where the Hamiltonian is conserved. https://en.wikipedia.org/wiki/Symplectic_integrator


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The general solution for $$\partial_t T(x,t)=\alpha\Delta T(x,t) \,\,\,\text{with}\,\,\, T(0,t)=T(L,t)=0$$ is $$u(x,t)=\sum_{k\in\mathbb{N}}a_k e^{-\alpha\lambda_kt}f_k(x),$$ where the functions $f_k$ are eigenfunctions of the La-Place-Operator to the eigenvalue $\lambda_k$ $$\Delta f_k=\lambda_k f_k,\,\,\,\,\,\,\,(1)$$ with boundary conditions $f_k(0,t)=f_k(...


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Let’s ignore the kinematic time dilation and estimate the gravitational time dilation. The dilation factor, approximately $1+\frac{GM}{rc^2}$ at large $r$, varies between $1+\frac{1}{6.5}\approx 1.15$ to $1+\frac{1}{15}\approx 1.07$. That’s about a 4% variation around the average, and it seems consistent with the slight visible wobble around the plot’s ...


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I see what they are trying to do here, but I'm not sure that the approach they use gives the actual solution to the equations as stated. Not that that matters, because it is only a simulation, and not an attempt for really quantify what is happening. The second equation actually describes the transport and generation of an inert tracer species (of low ...


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C++/OpenGL would be a great way to visualize all of those points. Perhaps put some of your data files up on GitHub, and I will take a peek. Are you at all familiar with OpenGL? That video link that you posted is for Vulkan, a competitor of OpenGL. Either way, you're looking at some graphics programming. I can help you with OpenGL code, and it will definitely ...


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