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annotate +time/+rk4/rungekutta_6.m @ 845:1e057b0f2fed feature/burgers1d

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Add RK6 with residual viscosity update and reduce computational effort of spatial scheme - Add RK6 with residual updates - Change the D2 operator for upwind schemes to one less computationally expensive.
author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Wed, 19 Sep 2018 16:32:05 +0200
parents 48b6fb693025
children
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1e057b0f2fed Add RK6 with residual viscosity update and reduce computational effort of spatial scheme
Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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1 % Takes one time step of size dt using the rungekutta method
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2 % starting from v_0 and where the function F(v,t) gives the
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3 % time derivatives.
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Vidar Stiernström <vidar.stiernstrom@it.uu.se>
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4 function v = rungekutta_6(v, t , dt, F)
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5 s = 7
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6 k = zeros(length(v),s)
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7 a = zeros(7,6);
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8 c = [0, 4/7, 5/7, 6/7, (5-sqrt(5))/10, (5+sqrt(5))/10, 1];
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9 b = [1/12, 0, 0, 0, 5/12, 5/12, 1/12];
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10 a = [
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11 0, 0, 0, 0, 0, 0;
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12 4/7, 0, 0, 0, 0, 0;
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13 115/112, -5/16, 0, 0, 0, 0;
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14 589/630, 5/18, -16/45, 0, 0, 0;
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15 229/1200 - 29/6000*sqrt(5), 119/240 - 187/1200*sqrt(5), -14/75 + 34/375*sqrt(5), -3/100*sqrt(5), 0, 0;
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16 71/2400 - 587/12000*sqrt(5), 187/480 - 391/2400*sqrt(5), -38/75 + 26/375*sqrt(5), 27/80 - 3/400*sqrt(5), (1+sqrt(5))/4, 0;
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17 -49/480 + 43/160*sqrt(5), -425/96 + 51/32*sqrt(5), 52/15 - 4/5*sqrt(5), -27/16 + 3/16*sqrt(5), 5/4 - 3/4*sqrt(5), 5/2 - 1/2*sqrt(5);
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18 ];
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19
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20 for i = 1:s
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21 u = v;
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22 for j = 1:i-1
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23 u = u + dt*a(i,j)*k(:,j);
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24 end
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25 k(:,i) = F(t+c(i)*dt,u);
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26 end
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27
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28 for i = 1:s
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29 v = v + dt*b(i)*k(:,i);
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30 end
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31 end