By Jichun Li
Short evaluation of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quickly evaluation of numerical tools for PDEsFinite distinction tools for Parabolic Equations creation Theoretical concerns: balance, consistence, and convergence 1-D parabolic equations2-D and three-D parabolic equationsNumerical examples with MATLAB codesFinite distinction equipment for Hyperbolic Equations IntroductionSome uncomplicated distinction schemes Dissipation and dispersion mistakes Extensions to conservation lawsThe second-order hyperbolic PDE. Read more...
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Additional info for Computational Partial Differential Equations Using MATLAB
35) can be obtained by the von Neumann technique. 35), we obtain the ampliﬁcation factor λ=1−4 t 1 1 t sin2 kx x − 4 sin2 ky y. 36) guarantees that |λ| ≤ 1 for all wavenumbers kx and ky . 35) is stable. 34): n δy2 (unr,s + un+1 un+1 1 δx2 (unr,s + un+1 r,s − ur,s r,s ) r,s ) = [ + ]. t 2 ( x)2 ( y)2 Denote μx = ( t x)2 , μy = ( t y)2 . 37) The CN scheme can be rewritten as 1 1 1 1 2 2 n (1 − μx δx2 − μy δy2 )un+1 r,s = (1 + μx δx + μy δy )ur,s . 37) can be easily obtained as n T Er,s = O(( t)2 + ( x)2 + ( y)2 ).
46) where for simplicity we assume that x = y = h. Denote μ = ht . By von Neumann stability analysis, the ampliﬁcation factor λ satisﬁes the equation 1 1 λ2 − 2λ + 1 = μ2 [−4λ sin2 kx x − 4λ sin2 ky y] 2 2 or 1 1 λ2 − 2[1 − 2μ2 (sin2 kx x + sin2 ky y)]λ + 1 = 0. 42). Similar ADI schemes as those of parabolic equations have been developed by many researchers (see  and references therein). 48) Finite Diﬀerence Methods for Hyperbolic Equations 49 if μ = 1. Furthermore, Fairweather and Mitchell  proved that the local truncation error is − 7 29 1 6 2 4 3 h μ [(μ − )(ux6 + uy6 ) + (μ4 − )(ux4 y2 + ux2 y4 )] = O(h6 ) 180 4 4 21 and the scheme is stability under the condition [1, Eq.
T Similarly, we can deﬁne the forward diﬀerence operator D+ in variable t as t v(x, t) = v(x, t + D+ t) − v(x, t). 1). , the numerical solution at node (xj , tn ) approximates the exact solution u(x, t) as (xj , tn ) gets close to the point (x, t), then we say that the scheme is convergent. 4 (Order of accuracy) If for a suﬃciently smooth solution u, T E(x, t) ≤ C[( t)p + ( x)q ], as t, x → 0, where p and q are the largest possible integers, then we say that the scheme has pth order of accuracy in t and qth order of accuracy in x.
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