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A Parallel Method for Heat Equation with Memory

A Parallel Method for Heat Equation with Memory. Kwon, Kiwoon and Sheen,Dongwoo. Dept. of Math. Seoul National University. Naturally Parallel Algorithms. Highly massive computing needs parallel computation. One of major naturally parallel algorithm is Domain decomposition method.

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A Parallel Method for Heat Equation with Memory

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  1. A Parallel Method for Heat Equation with Memory Kwon, Kiwoon and Sheen,Dongwoo Dept. of Math. Seoul National University

  2. Naturally Parallel Algorithms • Highly massive computing needs parallel computation • One of major naturally parallel algorithm is Domain decomposition method • Evolution equation is classically solved by time marching( stepping ) method, but • It is not parallelizable. • ButFrequency domain method is • Naturally parallelizable algorithm for evolution • equation

  3. Frequency domain method 1.Douglas Jr.,J E Santos,D Sheen : Wave with absorbing boundary condition 2.C-O Lee,J Lee, D Sheen, Y Yeom :heat equation 3.D Sheen,I H Sloan,V Thomee : heat equation(Fourier-Laplace transform) 4.C-O Lee,J Lee,D Sheen : linearized Navier-Stokes 5.K Kwon,D Sheen : heat equation with memory

  4. Classical heat equation 2. Unable to account formemory effects, which is prevalent in some materials 1. conservation law of energy 2. Fourier’s law Classical heat equation Drawbacks 1. A thermal disturbance at one point propagated instantly to everywhere of the body ( wave – inite speed)Classical heat equation : drawbacks

  5. Heat equation with memory • Coleman(64), Gurtin and Pipkin(68) : Replace Fourier’s law with equation with memory term Integro-differential equation: Applications 1.The transmission of heat pulses observed in liquid helium 2.Some dielectrics at low temperature

  6. K(s) is a constant  a wave equation • K(s) is a Dirac delta function  a heat equation • K(0) is finite • The speed of propagation is finite(wave) • K’(0) is divergent •  The speed of propagation is infinite • :The discontinuity is smoothed out(heat)

  7. Weak formulation • Original Problem where A is a symmetric positive definite operator The weak formulation:

  8. Positive Memory and Regularity • The memory Is called a positive memory if it satisfies for each • [Regularity]If is a positive Memory, then the solution satisfies is a positive memory

  9. Space-time domain • Fourier-Laplace Transform • Space-frequency domain

  10. Contour at a frequency domain • Is it possible to take a Fourier-Laplace transform at each point of a contour? • Is there a Space-Frequency domain solution at this frequency? (Avoid singular point!) • Is it possible to take a inverse Fourier-Laplace transform along the contour? • When any quadrature scheme is used, in which contour the order of convergence is good?

  11. Discretization in the space domain • (k-1)th degree finite element space and Ritz projection is used  When piecewise linear element is used

  12. where , Discretization in the frequency domain For It holds ,  where Point of the proof(SST) • Euler-MacLaurin formula • Spectral analysis • Semi group theory • Suitable choice of contour ,

  13. Fully discretization approximation Numerical Test(1D) Then the unique solution is

  14. Space Discretization Error

  15. Order of convergence Nx: space domain division number Nt: time domain division number Nz: frequency domain division number • Backward Euler: • Crank-Nicolson: • Frequency domain method: r:the regularity of right hand side • Strategy: • Choice of parameter • Approximation is bad if • T is too small or beta is too close to 0 or 1

  16. Two dimensional case

  17. References A study on inverse problems and numerical methods for partial differential equations,Ph.D thesis, Kiwoon Kwon, Dept. of math. Seoul National University, 2001,2.

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