Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6620 • Analysis of Numerical Methods II

Fall 2023 • Question 4

Question

Math 6620 • Analysis of Numerical Methods II Fall 2023 • Question 4

( Heat Equation and Stability of the Scheme). Consider the implicit in time, Backward Euler method for the solution of the heat equation: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} -10\frac {u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2} = f_m^{n+1}, \] \[ u_m^0 = g_m,\qquad m = 0,\pm 1,\pm 2,\ldots ,\qquad n = 0,1,\ldots ,[T/\Delta t]-1, \]

and investigate the stability of the scheme using the von Neumann analysis.

Raw LaTeX

( Heat Equation and Stability of the Scheme).
Consider the implicit in time, Backward Euler method for the solution of the
heat equation:
\[
\frac{u_m^{n+1}-u_m^n}{\Delta t}
-10\frac{u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2}
= f_m^{n+1},
\]
\[
u_m^0 = g_m,\qquad m = 0,\pm1,\pm2,\ldots,\qquad
n = 0,1,\ldots,[T/\Delta t]-1,
\]

and investigate the stability of the scheme using the von Neumann analysis.

Topic Tags

Absolute Stability and A-Stability Consistency, Stability, and Convergence Heat and Diffusion Equation Schemes Von Neumann Stability Analysis