Consider the unsteady heat diffusion (conduction) equation: = t u 2x As discussed in class, the general (weighted) implicit discretization scheme to numerically solve the above equation is: j u"+1-u" (u') n+1 - n+1 u' ,n+1 (^^x)= [ 5 (u}|| 2u;"' + u;;| ) + (15)(u%, 2u; +u%,)] where 0 1. This can be further written as: - n u;*' u; = y [5 (u; 2u;* +u;)+(15)(u; 2u; +u'})], _y= n+1 'j+1 ~~ (Ax) The modified equation for the above scheme - which contains only spatial derivatives - is presented in the lecture slides. Now do the following: (A) By examining the modified equation, obtain an expression for (as a function of y) for which the scheme becomes more accurate, i.e., [O(At), O(Ax)4]. (3 points) (B) Further show that if we choose the obtained in part (A), what specific value of y will make the scheme even more accurate, i.e., the truncation error of the scheme will become [O(At), O(Ax)]. (3 points) (C) Using the von Neumann stability analysis, show that the stability condition of this scheme is given by (1-2) 1/2. (7 points) (D) Find the stability requirement of this scheme using the matrix stability analysis method and compare your result with the result you obtained using the von Neumann method in part (C). (7 points)