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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2017 • Question 7

Question

Math 6610 • Analysis of Numerical Methods I Fall 2017 • Question 7

Heat Equation and Stability of the Scheme

Consider the implicit in time, Backward Euler method for the solution of the heat equation: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - \frac {u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2} = f_m^{n+1}, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1. \] 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:
\[
  \frac{u_m^{n+1}-u_m^n}{\Delta t}
  -
  \frac{u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2}
  =
  f_m^{n+1},
\]
\[
  u_m^0=g_m,\qquad
  m=0,\pm1,\pm2,\ldots,\qquad
  n=0,1,\ldots,[T/\Delta t]-1.
\]
and investigate the stability of the scheme using the von Neumann analysis.

Topic Tags

Absolute Stability and A-Stability Consistency, Stability, and Convergence Linear Programming and Optimization Von Neumann Stability Analysis