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>17:00 - 18:30 A General Theory of Local Preconditioning
COE�w�p�u����̂��m�点 �~�V�K����q��H�w�Ȃ̐��삳��ɐ��l�v�Z�@�ɂ��ču�����Ē����܂��B ���삳���ROE�@�Œm����AP.L.Roe�̌��q�����CFD�i���l���̗͊w�j�� ��b����������Ă��܂��B�F�l���U�����킹�̏㐥���Q�����������B ----------------------------------------------------------------------- �����F�Q�O�O�S�N�Q���Q�R���i���j �P�V�F�O�O�`�P�W�F�R�O �ꏊ�F���R�Ȋw�R���قU�K�Z�~�i�[���i�U�O�X�j �u���ҁF����T�� �~�V�K����w�q��H�w�� Title: A General Theory of Local Preconditioning Abstract: Local preconditioning, in the numerical solution of PDEs, is the attempt to accelerate convergence to the steady-state by altering the transient behavior of the PDEs. The technique has a particular aim of accelerating the error propagation associated with the physical wavespeeds of the PDEs. If these wavespeeds are very different, there will be error modes that take many more iterations than others to reach the boundary and expelled. The idea of local preconditioning is to multiply the time derivative in the PDE by a matrix (local preconditioning matrix) such that we equalize, as much as possible, the wavespeeds of the PDE to make all error modes propagate at the same rate, thereby accelerating the convergence. Although several such preconditioners have already been found for the Euler and the Navier-Stokes equations, none of them extends easily to other types of PDEs. Only recently, has a systematic way to derive optimal preconditioners for any two-dimensional PDEs been developed in the CFD group at the University of Michigan, based on elliptic/hyperbolic decomposition. This theory has made it possible to derive preconditioners for complex systems such as the magnetohydrodynamic(MHD) equations for which a preconditioner had not been available before. This talk will give an introduction to local preconditioning, describe the general theory, discuss its application to the MHD equations, and end with remark on further development to come.