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<title>MATKAIJ</title><body bgcolor="FFFFFF">
<div id="version" align=right><b>petsc-3.14.5 2021-03-03</b></div>
<div id="bugreport" align=right><a href="mailto:petsc-maint@mcs.anl.gov?subject=Typo or Error in Documentation &body=Please describe the typo or error in the documentation: petsc-3.14.5 v3.14.5 docs/manualpages/concepts/matkaij.html "><small>Report Typos and Errors</small></a></div>
<h2>MATKAIJ</h2>
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<LI><A HREF="../../../src/ksp/ksp/tutorials/ex74.c.html"><CONCEPT>ex74.c</CONCEPT></A>
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Solves the constant-coefficient 1D heat equation <BR>with an Implicit Runge-Kutta method using MatKAIJ. <BR>
<BR>
du d^2 u <BR>
-- = a ----- ; 0 <= x <= 1; <BR>
dt dx^2 <BR>
<BR>
with periodic boundary conditions <BR>
<BR>
2nd order central discretization in space: <BR>
<BR>
[ d^2 u ] u_{i+1} - 2u_i + u_{i-1} <BR>
[ ----- ] = ------------------------ <BR>
[ dx^2 ]i h^2 <BR>
<BR>
i = grid index; h = x_{i+1}-x_i (Uniform) <BR>
0 <= i < n h = 1.0/n <BR>
<BR>
Thus, <BR>
<BR>
du <BR>
-- = Ju; J = (a/h^2) tridiagonal(1,-2,1)_n <BR>
dt <BR>
<BR>
Implicit Runge-Kutta method: <BR>
<BR>
U^(k) = u^n + dt \\sum_i a_{ki} JU^{i} <BR>
u^{n+1} = u^n + dt \\sum_i b_i JU^{i} <BR>
<BR>
i = 1,...,s (s -> number of stages) <BR>
<BR>
At each time step, we solve <BR>
<BR>
[ 1 ] 1 <BR>
[ -- I \\otimes A^{-1} - J \\otimes I ] U = -- u^n \\otimes A^{-1} <BR>
[ dt ] dt <BR>
<BR>
where A is the Butcher tableaux of the implicit <BR>
Runge-Kutta method, <BR>
<BR>
with MATKAIJ and KSP. <BR>
<BR>
Available IRK Methods: <BR>
gauss n-stage Gauss method <BR>
<BR>
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