数学代写|偏微分方程代写partial difference equations代考|МATH4335

相信许多留学生对数学代考都不陌生,国外许多大学都引进了网课的学习模式。网课学业有利有弊,学生不需要到固定的教室学习,只需要登录相应的网站研讨线上课程即可。但也正是其便利性,线上课程的数量往往比正常课程多得多。留学生课业深重,时刻名贵,既要学习知识,又要结束多种类型的课堂作业,physics作业代写,物理代写,论文写作等;网课考试很大程度增加了他们的负担。所以,您要是有这方面的困扰,不要犹疑,订购myassignments-help代考渠道的数学代考服务,价格合理,给你前所未有的学习体会。

我们的数学代考服务适用于那些对课程结束没有掌握,或许没有满足的时刻结束网课的同学。高度匹配专业科目,按需结束您的网课考试、数学代写需求。担保买卖支持,100%退款保证,免费赠送Turnitin检测报告。myassignments-help的Math作业代写服务,是你留学路上忠实可靠的小帮手!


数学代写|偏微分方程代写partial difference equations代考|BANDED LINEAR SYSTEM SOLVERS

As discussed in the preceding section, Gaussian elimination has two major shortcomings. Both shortcomings stem from the fact that the entire coefficient matrix, including all zeroes, is stored and used. As we have seen in Chapter 2, coefficient matrices, arising out of discretization of any PDE, are sparse. This is true irrespective of the type of mesh used, as will be corroborated further in Chapter 7 . Unfortunately, a computer cannot distinguish between a zero and a nonzero automatically. It treats a zero as any other real number, and multiplications by zero are as time consuming as multiplications by nonzeroes. Therefore, a better approach would be to store only the nonzeroes and the locations in the matrix where they are situated. At the very least, this will reduce the memory usage significantly. In the example discussed in the last paragraph of the preceding section, for a 64,000-node mesh, the real number storage will reduce dramatically from $64,000^2$ to at most $64,000 \times 7$, assuming a hexahedral mesh is being used along with a second-order central difference scheme. Not storing the zeroes will also improve computational efficiency dramatically. This is because multiplications by zeroes will not be performed in the first place. For example, instead of performing a row times column multiplication involving $64,000^2$ long operations, such as in the computation of $\sum_{j=1}^K A_{i, j} \phi_j$, only $7 \times 64,000$ long operations will be performed since each row will have at most seven nonzero elements. To summarize, when it comes to solution of the linear system arising out of discretization of PDEs, using the sparseness of the coefficient matrix is warranted, as it can result in significant reduction in both memory usage and computational time.

A special class of sparse matrices arises out of discretization of a PDE if a structured mesh is used: banded matrices. This has already been alluded to in Chapter 2, although detailed discussion of this matter had been deferred to the present chapter. To elucidate the matrix structures represented by the equations derived in Chapter 2 , let us consider the solution of Eq. (2.1) subject to the boundary conditions shown by Eq. (2.2).

数学代写|偏微分方程代写partial difference equations代考|ITERATIVE SOLVERS

Direct solution to a linear system of equations produces the exact numerical solution to the PDE in question. Barring truncation and round-off errors, this solution has no other error when compared with the closed-form analytical solution to the same PDE. Therein lies its strength. Unfortunately, as shown in the preceding section, direct solution is prohibitive both from a memory usage and a computational efficiency standpoint. This leaves us with only one other alternative: solving the equations iteratively.

The philosophy of an iterative solution to a linear system is best explained by a simple example. Let us consider the following $3 \times 3$ system of equations:
$$
\begin{gathered}
5 x+2 y+2 z=9, \
2 x-6 y+3 z=-1, \
x+2 y+7 z=10 .
\end{gathered}
$$
The exact solution to the system is $[1,1,1]$. The Gaussian elimination method, by which this exact solution is obtained, is also known as an implicit method. An implicit method of solution is one where all unknowns are treated simultaneously during the solution procedure.

The solution to the above system may also be obtained using the following iterative procedure:

  1. Start with a guess for the solution.
  2. Solve Eq. (3.17a) for $x$ and update its value using old (guessed) values of $y$ and $z$.
  3. Solve Eq. (3.17b) for $y$ and update its value using old (guessed) values of $x$ and $z$.
  4. Solve Eq. (3.17c) for $z$ and update its value using old (guessed) values of $x$ and $y$.
  5. Replace old guess of solution by new solution.
  6. Repeat steps $2-5$ (i.e., iterate) until the solution has stopped changing beyond a certain number of significant digits, to be prescribed a priori.

In each of the steps 2 through 4 , two of the three unknowns are guessed, or treated explicitly, i.e., they are treated as known quantities and transposed to the right-hand side of the equation, while the unknowns are retained on the left-hand side. The fact that some terms in the equation receive explicit treatment implies that at each step, the solution obtained is only approximate, and therefore, iterations are necessary to correct the approximation. It will be shown later that how quickly the correct solution is approached, i.e., how many iterations are needed to arrive at the correct solution, depends on how many terms are treated explicitly, or the so-called degree of explicitness. Table $3.1$ shows how the values of $[x, y, z]$ change when the above iteration procedure is executed.

数学代写|偏微分方程代写partial difference equations代考|МATH4335

偏微分方程代考

数学代写|偏微分方程代写partial difference equations代考|BANDED LINEAR SYSTEM SOLVERS

如上一节所述,高斯消元法有两个主要缺点。这两个缺点都源于整个系数矩阵(包括全零)都被存储和使用的事实。正如我们在第 2 章中看到的,由任何 PDE 离散化产生的系数矩阵都是稀疏的。无论所使用的网格类型如何,这都是正确的,这将在第 7 章中进一步证实。不幸的是,计算机无法自动区分零和非零。它将零视为任何其他实数,乘以零与乘以非零一样耗时。因此,更好的方法是仅存储非零值和它们所在的矩阵中的位置。至少,这将显着减少内存使用量。64,0002至多64,000×7,假设六面体网格与二阶中心差分方案一起使用。不存储零也将显着提高计算效率。这是因为首先不会执行乘以零。例如,而不是执行行时间列乘法涉及64,0002长操作,例如在计算∑j=1ķ一个一世,jφj, 只要7×64,000将执行长操作,因为每行最多有七个非零元素。总而言之,当涉及到由 PDE 离散化产生的线性系统的求解时,使用系数矩阵的稀疏性是有保证的,因为它可以显着减少内存使用和计算时间。

如果使用结构化网格,则由 PDE 的离散化产生一类特殊的稀疏矩阵:带状矩阵。这已经在第 2 章中提到过,尽管对此问题的详细讨论已推迟到本章。为了阐明第 2 章推导出的方程所代表的矩阵结构,让我们考虑方程的解。(2.1) 受限于方程式所示的边界条件。(2.2)。

数学代写|偏微分方程代写partial difference equations代考|ITERATIVE SOLVERS

线性方程组的直接解法产生了所讨论 PDE 的精确数值解法。除截断和舍入误差外,该解与相同 PDE 的闭式解析解相比没有其他误差。它的力量就在于此。不幸的是,如上一节所示,从内存使用和计算效率的角度来看,直接解决方案是令人望而却步的。这给我们留下了另一种选择:迭代求解方程。

一个简单的例子可以很好地解释线性系统的迭代解决方案的原理。让我们考虑以下内容3×3方程组:

5X+2是+2和=9, 2X−6是+3和=−1, X+2是+7和=10.
系统的精确解是[1,1,1]. 获得该精确解的高斯消元法也称为隐式方法。隐式求解方法是在求解过程中同时处理所有未知数的方法。

上述系统的解决方案也可以使用以下迭代过程获得:

  1. 从对解决方案的猜测开始。
  2. 求解方程。(3.17a) 对于X并使用旧的(猜测的)值更新其值是和和.
  3. 求解方程。(3.17b) 对于是并使用旧的(猜测的)值更新其值X和和.
  4. 求解方程。(3.17c) 为和并使用旧的(猜测的)值更新其值X和是.
  5. 用新的解决方案替换旧的解决方案猜测。
  6. 重复步骤2−5(即迭代)直到解决方案停止变化超过一定数量的有效数字,这是先验规定的。

在步骤 2 到 4 的每一步中,三个未知数中的两个被猜测或明确处理,即,它们被视为已知量并转置到等式的右侧,而未知数保留在左侧 -手边。方程中的某些项得到显式处理这一事实意味着,在每一步,获得的解只是近似的,因此,需要迭代来纠正近似。稍后将显示接近正确解的速度,即需要多少次迭代才能达到正确解,取决于显式处理的项数,或所谓的显式程度。桌子3.1显示的值如何[X,是,和]执行上述迭代过程时更改。

数学代写|偏微分方程代写partial difference equations代考

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址,相关账户,以及课程名称,Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明,让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb,费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵),报价后价格觉得合适,可以先付一周的款,我们帮你试做,满意后再继续,遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款,全款,周付款,周付款一方面方便大家查阅自己的分数,一方面也方便大家资金周转,注意:每周固定周一时先预付下周的定金,不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

留学生代写覆盖学科?

代写学科覆盖Math数学,经济代写,金融,计算机,生物信息,统计Statistics,Financial Engineering,Mathematical Finance,Quantitative Finance,Management Information Systems,Business Analytics,Data Science等。代写编程语言包括Python代写、Physics作业代写、物理代写、R语言代写、R代写、Matlab代写、C++代做、Java代做等。

数学作业代写会暴露客户的私密信息吗?

我们myassignments-help为了客户的信息泄露,采用的软件都是专业的防追踪的软件,保证安全隐私,绝对保密。您在我们平台订购的任何网课服务以及相关收费标准,都是公开透明,不存在任何针对性收费及差异化服务,我们随时欢迎选购的留学生朋友监督我们的服务,提出Math作业代写、数学代写修改建议。我们保障每一位客户的隐私安全。

留学生代写提供什么服务?

我们提供英语国家如美国、加拿大、英国、澳洲、新西兰、新加坡等华人留学生论文作业代写、物理代写、essay润色精修、课业辅导及网课代修代写、Quiz,Exam协助、期刊论文发表等学术服务,myassignments-help拥有的专业Math作业代写写手皆是精英学识修为精湛;实战经验丰富的学哥学姐!为你解决一切学术烦恼!

物理代考靠谱吗?

靠谱的数学代考听起来简单,但实际上不好甄别。我们能做到的靠谱,是把客户的网课当成自己的网课;把客户的作业当成自己的作业;并将这样的理念传达到全职写手和freelancer的日常培养中,坚决辞退糊弄、不守时、抄袭的写手!这就是我们要做的靠谱!

数学代考下单流程

提早与客服交流,处理你心中的顾虑。操作下单,上传你的数学代考/论文代写要求。专家结束论文,准时交给,在此过程中可与专家随时交流。后续互动批改

付款操作:我们数学代考服务正常多种支付方法,包含paypal,visa,mastercard,支付宝,union pay。下单后与专家直接互动。

售后服务:论文结束后保证完美经过turnitin查看,在线客服全天候在线为您服务。如果你觉得有需求批改的当地能够免费批改,直至您对论文满意为止。如果上交给教师后有需求批改的当地,只需求告诉您的批改要求或教师的comments,专家会据此批改。

保密服务:不需求提供真实的数学代考名字和电话号码,请提供其他牢靠的联系方法。我们有自己的工作准则,不会泄露您的个人信息。

myassignments-help擅长领域包含但不是全部:

myassignments-help服务请添加我们官网的客服或者微信/QQ,我们的服务覆盖:Assignment代写、Business商科代写、CS代考、Economics经济学代写、Essay代写、Finance金融代写、Math数学代写、report代写、R语言代考、Statistics统计学代写、物理代考、作业代写、加拿大代考、加拿大统计代写、北美代写、北美作业代写、北美统计代考、商科Essay代写、商科代考、数学代考、数学代写、数学作业代写、physics作业代写、物理代写、数据分析代写、新西兰代写、澳洲Essay代写、澳洲代写、澳洲作业代写、澳洲统计代写、澳洲金融代写、留学生课业指导、经济代写、统计代写、统计作业代写、美国Essay代写、美国代考、美国数学代写、美国统计代写、英国Essay代写、英国代考、英国作业代写、英国数学代写、英国统计代写、英国金融代写、论文代写、金融代考、金融作业代写。

发表评论

您的电子邮箱地址不会被公开。 必填项已用*标注

Scroll to Top