     # Theta Elsevier

• View
24

• Category

## Documents

Embed Size (px)

### Text of Theta Elsevier

112d2dt2 + Ah

d2dt2uh(tn) + O(t4). (15)Knowing that uh(t) is solution of (2), we can replace the second order time derivative by the matrix Ah, givingD2tuh(tn) + Ah{uh(tn)} = t2

112

A2h uh(tn) + O(t4). (16)This expression shows that we obtain second order accuracy except for = 1/12, which cancels out the rst consis-tency error term, giving a fourth order scheme.In many situations, the time step t and the spatial discretization (via the spectral radius (Ah)) are related to thephysics and are considered as data for the numerical analyst. It is then reasonable to choose a numerical schemesuited to these parameters. A good criterion to compare the numerical schemes is the coecient of the rst term ofthe consistency error, which is given here by ( 1/12). As the -schemes give a degree of freedom, by adapting to the product t2(Ah), it is possible to decrease the consistency error while providing a stable scheme. For the-scheme, this optimization problem is straightforward: two cases arise, either t2(Ah) 6, in which case thevalue = 1/12 leads to an order 4 stable scheme (the relation (14) holds), or if t2(Ah) > 6, the choice of thatprovides a stable scheme and minimizes the consistency error is =14 1t2(Ah). (17)33. Construction of a family of fourth order implicit schemes3.1. Modied equation technique for the leap frog schemeThe modied equation technique, introduced in , enables to construct higher order schemes from the secondorder leap frog scheme. More precisely, the order 2p scheme is obtained by adding to the leap frog schemes termsthat compensate the (p 1) rst terms of the truncation error. This error reads, for uh(t) solution of (2):D2tuh(tn) + Ahuh(tn) = 2m=1t2m(2m+ 2)!d2m+2dt2m+2uh(tn). (18)Again, we can replace the second order time derivative by the matrix Ah:D2tuh(tn) + Ahuh(tn) = 2m=1(1)p+1 t2m(2m + 2)!Am+1h uh(tn). (19)Truncating the series up to the rst term, and approaching uh(tn) with unh gives the fourth order scheme:D2tunh + Ahunh t212 A2hunh = 0. (20)The stability condition is deduced from energy arguments and readst2 B00(Ah), (21)where B00 = 12.Using an H orner algorithm, this scheme is two times more expensive than the original scheme. This over cost can becompensated by adding stabilization terms that allow to increase the time step as explained in  and .3.2. Modied equation technique for the -schemeWe now use the ideas of the modied equation technique applied to the -scheme (instead of the leap frog scheme,i.e = 0). Let uh(t) be solution of

Recommended Documents