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AMATH 581 ( c○J. N. Kutz) 39ωmnt=2 Δtω mnt= Δtt=0ω mnFigure 13: Discretization stencil resulting from center-differencing of theadvection-diffusion equations. Note that for explicit stepping schemes the futuresolution only depends upon the present. Thus we are not required to solvealargelinearsystemofequations.2.2 Direct solution methods for Ax=bAcentralconcerninalmostanycomputationalstrategyisafast and efficientcomputational method for achieving a solution of a large system of equationsAx = b. Intryingtorenderacomputationtractable,itiscrucialtominimizethe operations it takes in solving such a system. There are a variety of directmethods for solving Ax = b: Gaussian elimination, LU decomposition, andinverting the matrix A. Inadditiontothesedirectmethods,iterativeschemescan also provide efficient solution techniques. Some basic iterative schemes willbe discussed in what follows.The standard beginning to discussions of solution techniques for Ax = binvolves Gaussian elimination. We will consider a very simple example of a3×3systeminordertounderstandtheoperationcountandnumerical procedure