Numberical Solution of Algebraic Systems Notes

Numberical Solution of Algebraic Systems Notes - AIMS...

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AIMS Lecture Notes 2006 PeterJ .O lver 11. Numerical Solution of the Heat and Wave Equations In this part, we study numerical solution methodss for the two most important equa- tions of one-dimensional continuum dynamics. The heat equation models the diFusion of thermal energy in a body; here, we treat the case of a one-dimensional bar. The wave equation describes vibrations and waves in continuous media, including sound waves, wa- ter waves, elastic waves, electromagnetic waves, and so on. ±or simplicity, we restrict our attention to the case of waves in a one-dimensional medium, e.g., a string, bar, or column of air. We begin with a general discussion of ²nite diFerence formulae for numerically ap- proximating derivatives of functions. The basic fnite diFerence scheme is obtained by replacing the derivatives in the equation by the appropriate numerical diFerentiation for- mulae. However, there is no guarantee that the resulting numerical scheme will accurately approximate the true solution, and further analysis is required to elicit bona ²de, conver- gent numerical algorithms. In dynamical problems, the ²nite diFerence schemes replace the partial diFerential equation by an iterative linear matrix system, and the analysis of convergence relies on the methods covered in Section 7.1. We will only introduce the most basic algorithms, leaving more sophisticated variations and extensions to a more thorough treatment, which can be found in numerical analysis texts, e.g., [ 5 , 7 , 28 ]. 11.1. Finite Diferences. In general, to approximate the derivative of a function at a point, say f 0 ( x )or f 0 ( x ), one constructs a suitable combination of sampled function values at nearby points. The underlying formalism used to construct these approximation formulae is known as the calculus o± fnite diFerences . Its development has a long and influential history, dating back to Newton. The resulting fnite diFerence numerical methods for solving diFerential equations have extremely broad applicability, and can, with proper care, be adapted to most problems that arise in mathematics and its many applications. The simplest ²nite diFerence approximation is the ordinary diFerence quotient u ( x + h ) u ( x ) h u 0 ( x ) , (11 . 1) 4/20/07 186 c ± 2006 Peter J. Olver
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One-Sided Diference Central Diference Figure 11.1. Finite Diference Approximations. used to approximate the ±rst derivative o² the ²unction u ( x ). Indeed, i² u is diferentiable at x ,th en u 0 ( x ) is, by de±nition, the limit, as h 0 o² the ±nite diference quotients. Geometrically, the diference quotient equals the slope o² the secant line through the two points ( x, u ( x ) ) and ( x + h, u ( x + h ) ) on the graph o² the ²unction. For small h i s should be a reasonably good approximation to the slope o² the tangent line, u 0 ( x ), as illustrated in the ±rst picture in Figure 11.1.
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Numberical Solution of Algebraic Systems Notes - AIMS...

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