本文選題：奇異退化擴散反應方程 + 自適應方法��； 參考：《寧夏大學》2016年碩士論文

【摘要】：研究奇異退化擴散反應方程解的爆破時間和爆破空間位置,具有重要的理論意義和實際應用價值.對此類方程的求解已經(jīng)發(fā)展了許多數(shù)值算法.就目前的研究現(xiàn)狀而言,主要是基于均勻網(wǎng)格上實施的,或者是基于低精度格式的自適應算法實施的,而關于奇異退化擴散反應方程的高精度緊致差分格式及網(wǎng)格自適應方法的研究非常少見.利用非均勻網(wǎng)格上的高精度緊致差分格式求解梯度變化較大的問題時,其計算精度相對于均勻網(wǎng)格的高精度緊致差分格式具有明顯的優(yōu)勢.本文首先利用截斷誤差余項修正法建立了一維奇異退化擴散反應方程在非均勻網(wǎng)格上的高階緊致差分格式,根據(jù)推導過程可知其時間具有二階精度,空間具有三階至四階精度,網(wǎng)格是兩層三點模版,可直接利用追趕法進行求解.在爆破點處溫度關于時間的導數(shù)會產(chǎn)生大幅度的跳躍,本文利用等分布原理建立了時間和空間的網(wǎng)格自適應方法.由于網(wǎng)格產(chǎn)生了移動,所以利用線率對新舊網(wǎng)格上的函數(shù)值進行傳遞.并將方法從一維問題推廣到高維問題.最后通過具有精確解的數(shù)值算例驗證了本文方法的精確性和穩(wěn)定性,再應用本文方法對爆破問題進行數(shù)值模擬,并與已有的數(shù)值結果進行比較,發(fā)現(xiàn)本文計算結果與文獻中的數(shù)值結果相吻合.
[Abstract]:It is of great theoretical significance and practical application value to study the blasting time and space position of the solution of singular degenerate diffusion reaction equation. Many numerical algorithms have been developed for solving such equations. As far as the current research situation is concerned, it is mainly based on uniform mesh or adaptive algorithm based on low precision format. However, there are few studies on compact difference schemes and mesh adaptive methods for singular degenerate diffusion equations. When the high precision compact difference scheme on non-uniform mesh is used to solve the problem with large gradient variation, its computational accuracy is obviously superior to that of the high precision compact difference scheme of uniform grid. In this paper, first of all, by using the truncation error remainder correction method, the high order compact difference scheme of one-dimensional singular degenerate diffusion reaction equations on non-uniform grids is established. According to the derivation process, the time has second-order accuracy and the space has third-to fourth-order accuracy. The grid is a two-layer three-point template, which can be solved directly by the catch-up method. The derivative of temperature on time at the blasting point will produce a large jump. In this paper, the mesh adaptive method of time and space is established by using the principle of equal distribution. Because the grid is moving, the line rate is used to transfer the function values on the new and old grids. The method is extended from one-dimensional problem to high-dimensional problem. Finally, the accuracy and stability of the proposed method are verified by a numerical example with exact solution. The numerical simulation of the blasting problem is carried out by using the proposed method, and the numerical results are compared with the existing numerical results. It is found that the calculated results in this paper are in agreement with the numerical results in the literature.
【學位授予單位】：寧夏大學
【學位級別】：碩士
【學位授予年份】：2016
【分類號】：O241.82

【參考文獻】

相關期刊論文 前2條

1 葛永斌;田振夫;吳文權;;高維熱傳導方程的高精度交替方向隱式方法[J];上海理工大學學報;2007年01期

2 葛永斌,田振夫,吳文權;二維拋物型方程的高精度多重網(wǎng)格解法[J];應用數(shù)學;2003年02期

，

本文編號：2114990