劉 倩, 胡勁松, 林雪梅

(西華大學(xué)數(shù)學(xué)與計(jì)算機(jī)學(xué)院,四川成都610039)

在研究弱非線性離子聲波和空間帶電波的傳播時(shí),文獻(xiàn)[1]提出了廣義對(duì)稱正則長(zhǎng)波(SRLW)方程:

方程(1)～(2)也出現(xiàn)在許多其他數(shù)學(xué)物理研究領(lǐng)域[2-4].關(guān)于SRLW方程的定解問(wèn)題的適定性及數(shù)值方法的研究也引起了廣泛關(guān)注[5-12].在實(shí)際問(wèn)題中,粘性阻尼是不可避免的,而且與色散一樣起著十分重要的作用.本文考慮如下一類帶有阻尼項(xiàng)的耗散廣義SRLW方程的初邊值問(wèn)題:

其中,p≥2為正整數(shù),υ>0是耗散系數(shù),γ>0是阻尼系數(shù).

在考慮耗散時(shí),方程(3)～(4)是反映非線性離子聲波運(yùn)動(dòng)本質(zhì)現(xiàn)象的合理模型[13],文獻(xiàn)[13-17]分別討論了方程(3)～(4)的解的適定性和整體存在唯一性以及解的長(zhǎng)時(shí)間性態(tài)等,但其解析解很難求出.于是,研究初邊值問(wèn)題(3)～(6)的數(shù)值解就很有理論價(jià)值和應(yīng)用價(jià)值.文獻(xiàn)[18-19]分別討論了初邊值問(wèn)題(3)～(6)在p=2時(shí)的差分近似解,本文考慮更一般的情形,構(gòu)造了問(wèn)題(3)～(6)的一個(gè)三層線性有限差分格式,數(shù)值計(jì)算時(shí)不需要迭代,時(shí)間比較節(jié)省,討了其差分解的先驗(yàn)估計(jì),分析了該格式的二階收斂性和無(wú)條件穩(wěn)定性,同時(shí)給出數(shù)值算例來(lái)說(shuō)明該格式的有效性.

1 線性差分格式及差分解的估計(jì)

2 差分格式的收斂性與穩(wěn)定性

表1 p=3時(shí),在幾個(gè)不同時(shí)刻的誤差(τ=h)Table 1 The errors of numerical solutions at various time step with τ =h for p=3

表2 p=5時(shí),在幾個(gè)不同時(shí)刻的誤差(τ=h)Table 2 The errors of numerical solutions at various time step with τ =h for p=5

3 數(shù)值實(shí)驗(yàn)

在t=0時(shí),由于阻尼還沒(méi)有作用,耗散還沒(méi)有產(chǎn)生,所以在數(shù)值實(shí)驗(yàn)中,把問(wèn)題(3)～(6)中的初值函數(shù)取為廣義 SRLW方程(1)、(2)的初值函數(shù)[5](t=0時(shí)):

對(duì)初邊值問(wèn)題(3)～(6)考慮p=3和p=5這2種情況進(jìn)行數(shù)值實(shí)驗(yàn).固定-xL=xR=20,T=1.0,取υ=γ=1.由于方程(3)～(4)的精確解并不知道,用類似文獻(xiàn)[18-19]的誤差估計(jì)方法,將細(xì)網(wǎng)格(h=τ=1/160)上的數(shù)值解作為精確解來(lái)估計(jì)誤差,就τ和h的不同取值時(shí),幾個(gè)不同時(shí)刻的l∞誤差見(jiàn)表1和2.

從數(shù)值算例看出,本文的格式明顯具有二階精度,而且是一個(gè)線性格式,計(jì)算時(shí)間也比較節(jié)約.所以本文的格式是實(shí)用而可靠的.

致謝西華大學(xué)研究生創(chuàng)新基金(YCJJ201311)對(duì)本文給予了資助,謹(jǐn)致謝意.

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