A Parallel Scheme For Solving Hyperbolic Equations With Legendre Pseudo-spectral Method

In this paper we consider the initial-boundary value problem offollowing hyperbolic equations and their parallel scheme.where are given smooth functions. satisfies thefollowing conditions:The main purpose of this paper is to set up the semi-discreteapproximation to the equation with pseudo-spectral method andanalyze their convergence property. Then we divide the originaldomain into pieces and solve on each subdomains with semi-discreteequations, and we study the parallel scheme .At last, we provide anumerical example to illustrate our results.First, we introduce the pseudo-spectral method and the parallelmethod.The Legendre pseudo-spectral method is the method which setsup the semi-discrete scheme for the problem by choosing Legendre -Gauss-Lobatto quadrature crunodes as the collocation points. It hasgood approximation property,that makes the pseudo-spectral to be anumerical method with high precision.Parallel method and pseudo-spectral have either advantages, westudy them together, it will make calculate more simple andconvenient.We first set up a semi-discrete scheme of equations (1).We use the pseudo-spectral method to deal with the spacevariable x and get the following semi-discrete problem:Find a u c ∈C 1 ((0, T ];S N?1( I)), such that101 0( 1, ) ( 1) ( 1, ) ( 1)( ( 1, ) ( )) ( 1, ),( , ) ( ) ( , ) ( , ), 1 1, (2)( ,0) ( )( ), 0 1.ct cx cct j j cx j jc j N ju t b u t b u t t f tu x t b x u x t f x t j Nu x P u x j Nω ?ψ???? ? ++ ? ? =+ ? ?≤ ≤? ?= ??? = ≤ ≤ ?The collocation points used are the Gauss-Lobatto type related to theLegendre polynomials. x0 = ? 1,x( j1 ≤ j ≤N-1) is the root ofLN′ ,where LN ( x) is the Legendre polynomial with N degree.02ω = N ( N+ 1) is the integration coefficient.Then we obtain the error estimate for equations (2).Theorem 1 if u and u care the solutions of (1) and (2) ,2 1u , f ∈ L∞ ( H σ ( I )), u 0∈ H σ ( I ), f t ∈ L ( H σ ( I )), u t∈ L∞ ( H σ?( I))IL2 ( H σ ( I)) , u tt∈L2 ( H σ ?1(I)), we can prove2 21 2 2 11 2( ) ( ) ( )0 ( ) ( ) ( ) ( ){ }{ }c L L L H tL HL H t L H t L H ttL Hu u cN f fcN u u u u uσ σσ σ σσσσ∞ ∞∞ ∞ ???? ≤ ++ + + + + .σ?where ( ) ([0, ];( )) , 2,Ls H Ls T H I sσ σ? = ? = ∞ .Then we divide the original domain into pieces. we set up thesimilar semi-discrete Legendre pseudo-spectral equations on eachsubdomains , and set up a parallel scheme.Let I = ( ? 1,1),decomposeI into I1 , I 2 , K , I S , I s = ( xs ?1 , xs ), s=1, K , S.let D = (0, T],decompose D intoD1 , D2 , L , DM , Dm =(t m ?1, tm],m = 1, L ,M .wherex0 =?1 , xS= 1,t 0 = 0, t M= T.let Ω sm = I s × Dm, noteu Ω sm = usm.In this way we decompose Ω = ( ?1 ,1) × ( 0,T] intosubdomains.S ×MMake a transform of x: [ ]( )1 11 ( ) 1( ),2 2sx = x s ? x s ? x + xs + x s?x∈ ?1,1 .that( )1 11 ( ) 1( ), 0, , 1, 1, ,2 2sx j = x s ? x s ? x j + xs ? xs ?j = L N ? s =L SWhere x j, j = 0, L ,N ?1, are the N collocation points in the originaldomain Ω.We find the equation (2) is equivalent to:find a u cs , m∈C 1 (( t m ?1 , t m ];S N?1( I)),so that( ) ( ) ( ) 1 ( ) ( ) ( )0 0 0 0 0 0 0( ) ( ) ( ) ( )( ) ( )1 1 0( , ) ( ) ( , ) ( )( ( , ) ( )) ( , ),( , ) ( ) ( , ) ( , ), 1 1,( , ) ( )( ), 0 1.sm s s sm s s sm s sm sct cx csm s s sm s sct j j cx j jsm s sm sc j m N ju x t b x u x t b x u x t t f x tu x t b x u x t f x t j Nu x t P u x j Nω ?ψ? ????? ++ += ≤ ≤? ?=??? = ≤ ≤ ?where( 1)1( ), 1,()( , ), 2, ,sms mc st stu x t sψψ = ??? ? ? ==L S,( )( )1 01 ( 1) ( )1( )( ), 1,( , )( , ), 2, ,ssm sN jc j m s m sc j mP u x mu x tu x t m?? ??= ????? ==L M .The relation of the solutions is as Figure (1).It illuminates that thesolutions on Ω sm at most depend on the values on Ω s ( m?1)and Ω ( s ?1)m.The sequence of the parallel scheme is as figure(2).Ωs (m ?1)ΩsmΩ(s ? 1)mΩ 11Ω 12Ω 13Ω 14Ω15Ω21Ω22Ω23Ω24Ω25Ω 31Ω 32Ω33Ω 34Ω35Ω 41Ω 42Ω 43Ω 44Ω45Figure (1) Figure (2)At the end of the paper we give a numerical example to illustrateour theory. The numerical results fit to the exact solution very well.