petsc-3.12.4 2020-02-04
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## MATKAIJ

- <CONCEPT>ex74.c</CONCEPT> Solves the constant-coefficient 1D Heat equation with an Implicit
  Runge-Kutta method using MatKAIJ.
  du d^2 u
  -- = a ----- ; 0 <= x <= 1;
  dt dx^2
  with periodic boundary conditions
  2nd order central discretization in space:
  [ d^2 u ] u_{i+1} - 2u_i + u_{i-1}
  [ ----- ] = ------------------------
  [ dx^2 ]i h^2
  i = grid index; h = x_{i+1}-x_i (Uniform)
  0 <= i < n h = 1.0/n
  Thus,
  du
  -- = Ju; J = (a/h^2) tridiagonal(1,-2,1)_n
  dt
  Implicit Runge-Kutta method:
  U^(k) = u^n + dt \\sum_i a_{ki} JU^{i}
  u^{n+1} = u^n + dt \\sum_i b_i JU^{i}
  i = 1,...,s (s -> number of stages)
  At each time step, we solve
  [ 1 ] 1
  [ -- I \\otimes A^{-1} - J \\otimes I ] U = -- u^n \\otimes A^{-1}
  [ dt ] dt
  where A is the Butcher tableaux of the implicit
  Runge-Kutta method,
  with MATKAIJ and KSP.
  Available IRK Methods:
  gauss n-stage Gauss method