Thank you for your helpful answer to my question.
Is it true that the general heat diffusion PDE cannot be solved in real time?
In reality
We propose that the most likely answer is unfortunately yes, it is impossible to solve the general heat diffusion PDE in real time.
I know the question and its answer are very difficult but not ambiguous or unclear.
The question itself is of utmost importance and I have offered my description of a new numerical technical answer for Laplace's and Poisson's solution in many posts in Researshgate and IJISRT Revew. Please read the following references as an example,
1-A rigorous experimental technique for measuring the thermal diffusivity of metals
2-In the heat diffusion conduction equation, how to extend the validity of the Dirichlet boundary conditions to more than one dimensional geometric space.
3-Why it is impossible to solve the heat diffusion equation as a function of time in 2D and 3D situations.
Thank you for your helpful answer but,
(I don't understand your question. Are you asking if there is a computer and/or algorithm fast enough to obtain a mathematical solution for a process as it occurs (i.e., "real time")? I have solved countless examples of heat transfer and diffusion using Laplace's PDE. We usually separate heat transfer (movement of energy and with or without mass) from diffusion (movement of mass with or without energy). Here's a link to several examples.)
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There must be a missunderstanding .
I know you unfortunately didn't understand the question or its answer, but believe me, it's true.
Below is the general form of the heat diffusion partial differential equationwe are investigating, except we use D for the thermal diffusion coefficientinstead of Alpha,d / dt (partial) U (x,y,z,t) = D Nabla2 U (x,y,z,t) + S (x,y,z,t) .. . . . . . . . .(1)Subject to the boundary conditions of Dirichlet B C on the limits of thedomain of U and the initial conditions IC of U at t=0 that is U ( x,y,z,0).Where,U(x,y,z,t) represents temperature energy density.U(x,y,z,t)=K .T(x,y,z,t) , T in degrees Kelvin and KB is Boltzmannconstant.S(x,y,z,t) is the heat energy density source/sink term at thecorresponding free nodes in the considered 2D or 3D domain.
IF you know any answer to Eq 1 in real time please letr me know ! (PDF) In the heat diffusionconduction equation, how to extend the validity of the Dirichlet boundary conditions to more than one dimensional geometric space.. Available from: https://www.researchgate.net/publication/361408922_In_the_heat_diffusionconduction_equation_how_to_extend_the_validity_of_the_Dirichlet_boundary_conditions_to_more_than_one_dimensional_geometric_space [accessed Sep 01 2022].
Solving an equation in real time is about the speed with which it can be done. Also, the context would be important. Typically heat flows are relatively slow -- seconds rather than milli- or micro-seconds or less (as might be appropriate for various wave systems, esp. electromagnetic ones). Weather problems include heat/diffusion effects & equations, and need to be solved over hours to provide "real time" predictions.
That being said, if the problem is say essentially two-dimensional (plates, perhaps), then the solvers can be quite fast. Large 3-D problems with complex geometries would be harder, but could probably be done with some clever computational methods, such as
* precomputing the mass/stiffness matrices (perhaps using a lumped mass approximation),
* solving the equations coming from an (implicit) ODE solver for the finite element/difference/volume equations using iterative methods and good starting approximations (say, a low order predictor to feed into the implicit solver).
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