The heat equation Fourier and Crank-Nicolsen Repetition and additional notes

Solutions of the heat equation smooths out Suppose one has a function u that describes the temperature at a given location (x, y, z). This function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time. The rate of change of u is proportional to the "curvature" of u. Thus, the sharper the corner, the faster it is rounded off. Over time, the tendency is for peaks to be eroded, and valleys filled in. If u is linear in space (or has a constant gradient) at a given point, then u has reached steady-state and is unchanging at this point (assuming a constant thermal conductivity).

Wikipedia on Fourier series (2016) Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles.

Fourier: Analytical theory of heat (1822)

Heat and diffusion equations correspond to conservation of energy and mass respectively

There are many applications The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation. In financial mathematics it is used to solve the Black–Scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes. …. The wave equation can be approximately solved by the so-called parabolic approximation which gives a similar equation.

Medisinsk ultralyd 2010 ?? (written in year 2000) http://www.cs.unc.edu/~us/

Second harmonic tissue imaging RF-signal from aortic valve Source: Hans Torp 1997 Second harmonic tissue imaging RF-signal from aortic valve Fundamental 2. harmonisk

http://www.ntnu.edu/isb/ultrasound/abersim T. Varslot and G. Taraldsen, Computer Simulation of Forward Wave Propagation in Soft Tissue, IEEE Trans. on UFFC, 52(9):1473-1482, 2005. Abersim© 2.0 Users Manual (pdf).

Fourier transform and convolution solution

Finally …. Crank-Nicolson = trapezoidal rule Numerical solution Finally …. Crank-Nicolson =  trapezoidal rule

Wikipedia (2016) on Crank-Nicolson For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable.[3] However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step Δt times the thermal diffusivity to the square of space step, Δx2, is large (typically larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations.

ODE solvers for 1d heat-equation relatives

Simplified heat equation

Physical interpretation and numerics

Stencils for direct Euler and Crank-Nicolson

Crank-Nicolson for simplified heat equation

We have to perform 4 times as many steps as with the Crank–Nicolson

The End