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五代數(shù)方程的求解(參考版)

2025-07-20 12:48本頁面
  

【正文】 ? Peric《 書 》 Chapter 5 全部。但不能處理高階導(dǎo)數(shù)項。*eq(i+1) +eq(i) results in: 40 Comparison of count for convergence ? On 2D Poisson equation, k*k grid, n=k2, unknown Method Cost Gaussian elimination O(n3) GS O(n2logn) CG O() FFT/Cyclic reduction O(nlogn) Multigrid O(n) optimal 41 選擇 solver ? MG+SIP or MG +GS ICCG SIP ADI GS ? GMRES+MG ? 沒有 MG時, ICCGSIP ADI GS 42 其他迭代法 coupled equations (system of nonlinear equations) ? Simultaneous approach: All equations are considered part of a single system. ? Sequential approach: Each equation is solved for its dominant variable, treating the other as known, and one iterates through the equations until the solution of the coupled system is obtained. ? Iterations performed on each equation are called inner iteration. ? In order to obtain a solution which satisfy all equations, the coefficient matrices and source vector must be updated after each cycle ad the process repeated. The cycle are called outer iteration. 43 Sequential solution: UnderRelaxation On the nth iteration the equation for generic variable is Patankar 1980對 SIMPLE采用 , 穩(wěn)定求解,但可能降低收斂率 時間相關(guān)法就是一種松馳法。在最粗網(wǎng)格上精確求解修正方程??梢栽诖志W(wǎng)格上消除這些低頻分量。 28 Conjugate gradient methods ? 線性代數(shù)方程和極小化: 對于對稱正定矩陣 A,求解 ?共軛 : 0212211021??????PPPPAAFFPP是共軛的:關(guān)于矩陣如果這兩個方向意一個方向做極小化,我們只需要針對其中任:平面內(nèi)極小化假設(shè)在???等價于找到 , 使 F極小 化 : i?29 Conjugate gradient methods (cont) ? 最速下降法:收斂慢,搜索方向可能重復(fù) ? 共軛梯度法:新的搜索方向要求和過去所有的搜索方向共軛 n*n矩陣, n次搜索就可以收斂 CG的收斂速度依賴于 A的條件
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