一類地下水污染物運(yùn)移問題的有限體積元方法

駱小芳, 朱泉涌, 林銀河*

(1. 南京師范大學(xué) 數(shù)學(xué)科學(xué)學(xué)院, 江蘇 南京 210023; 2. 麗水學(xué)院 數(shù)學(xué)系, 浙江 麗水 323000)

地下水作為水資源的重要組成部分,就水體污染而言,地下水的污染與地表水的污染相比更具有隱蔽性和難以逆轉(zhuǎn)性.地下水一旦受污染,便很難治理及恢復(fù).由于地下水污染流體動力系統(tǒng)的復(fù)雜性,傳統(tǒng)的實(shí)驗(yàn)研究方法花費(fèi)大、周期長、靈活性差且難以重復(fù)實(shí)驗(yàn).因此,研究建立在數(shù)學(xué)物理方法基礎(chǔ)上的地下水污染數(shù)值方法具有重要的理論和應(yīng)用價值[1-8].本文研究地下水污染模型問題[9-13]

(1)

的有限體積元解,其中,R為阻滯因子,s表示地下水污染物的濃度,k表示地下水實(shí)際流動速度,d表示擴(kuò)散系數(shù),λ表示衰減系數(shù),s0(x)為已知的光滑函數(shù),D=[0,L].方程(1)在地下水污染物運(yùn)移問題中具有深刻的物理背景,并且有著廣泛的應(yīng)用[14-17].

為方便討論，R取為1，假設(shè)問題(1)的解存在唯一且有必要的光滑性,并滿足下列條件:

|k(x,q1)-k(x,q2)|≤K1|q1-q2|;

(IV) ?x∈D,有0

下文中用K和ε分別表示一般的正常數(shù)和充分小的正常數(shù),它們在不同的地方可以表示不同的值.此外,記

1 有限體積元離散方法

選取試探函數(shù)空間Ch為相應(yīng)于剖分Th的拉格朗日型二次有限元空間,則整節(jié)點(diǎn)xi與半整節(jié)點(diǎn)xi-1/2的基函數(shù)分別為

而任意sh∈Ch可唯一表示為

φi(x)+si-1/2φi-1/2(x)],

其中si=sh(xi,t),si-1/2=sh(xi-1/2,t).

則任一wh∈Wh可表示為

其中,wj=wh(xj),wj-1/2=wh(xj-1/2).

?s∈Ch.

對?sh∈Ch,引入下面的離散范數(shù)

|sh|1,h=

引理1[9]對任意的sh∈Ch,|sh|0,h、|sh|1,h分別與|sh|0、|sh|1等價,即存在與h無關(guān)的正常數(shù)c1、c2、c3、c4,使得

c1|sh|0,h≤|sh|0≤c2|sh|0,h,

c3|sh|1,h≤|sh|1≤c4|sh|1,h.

引理2[9]下列結(jié)論成立:

引理3[9]存在正常數(shù)h0、α和M,使得當(dāng)0

c5‖sh‖1≤|||sh|||1≤c6‖sh‖1,

?sh∈Ch.

引入記號

sh=si-1(2ξ-1)(ξ-1)+

4si-1/2ξ(1-ξ)+siξ(2ξ-1),

(3)

(4)

即

2 誤差估計

記ξ=sh-Πhs,η=s-Πhs,則sh-s=ξ-η.由(2)和(4)式得誤差方程

(6)

由引理3得

下面對(6)式右端各項分別進(jìn)行估計,運(yùn)用ε不等式,容易得到

但

故

由

而

故

則

由

M(△t)2,

則

由

故

以上各式聯(lián)立，可得

(7)

綜上所述可得如下定理.

定理1假設(shè)s是問題(1)的解,sh為全離散有限體積二次元格式(4)的解,當(dāng)h與△t充分小時,有以下的誤差估計成立

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