Math 6610 • Analysis of Numerical Methods I

Heat Equation Consider the variable coefficient diffusion equation \[ v_t=(\beta v_x)_x,\qquad 0<x<1,\quad t>0 \] with Dirichlet boundary conditions \[ v(0,t)=0.0,\qquad v(1,t)=0.0 \] and initial data \(v(x,0)=f(x)\). Assume that \(\beta (x)\ge \beta _0>0\), and that \(\beta (x)\) is smooth. For any fixed value of \(\theta \in [0,1]\), consider the \(\theta \)-scheme for this problem: \[ \begin {aligned} \frac {u_j^{n+1}-u_j^n}{k} =\frac {1}{h^2}\{& (1-\theta )(\beta _{j-1/2}u_{j-1}^{n+1} -(\beta _{j-1/2}+\beta _{j+1/2})u_j^{n+1} +\beta _{j+1/2}u_{j+1}^{n+1})\\ &+\theta (\beta _{j-1/2}u_{j-1}^n -(\beta _{j-1/2}+\beta _{j+1/2})u_j^n +\beta _{j+1/2}u_{j+1}^n)\}. \end {aligned} \] with appropriate treatment of the boundary conditions in the relevant equations. Here, \(h\) is the spacestep, \((J+1)h=1\), \(k\) is the timestep, and \(\beta _{j+1/2}=\beta (x_{j+1/2})\), where \(x_{j+1/2}=(j+1/2)h\). Let \(M\) be the symmetric \(J\times J\) matrix defined by \[ \setlength {\arraycolsep }{3pt} M= \begin {pmatrix} -(\beta _{1/2}+\beta _{3/2}) & \beta _{3/2} & \cdots & \cdots & \cdots & \cdots \\ \beta _{3/2} & -(\beta _{3/2}+\beta _{5/2}) & \beta _{5/2} & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \beta _{J-3/2} & -(\beta _{J-3/2}+\beta _{J-1/2}) & \beta _{J-1/2}\\ \cdots & \cdots & \cdots & \cdots & \beta _{J-1/2} & -(\beta _{J-1/2}+\beta _{J+1/2}) \end {pmatrix}. \] Analyze the stability of this scheme in terms of \(\theta \), \(\frac {k}{h^2}\), and the eigenvalues of \(M\). Are there values of \(\theta \) for which the scheme is stable for all choices of \(\frac {k}{h^2}>0\)? (Note that answering this question should not require lots of writing.)