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2 2 A General Algorith for Local Error Control in the RKrGL Method Justin S. C. Prentice Departent of Applied Matheatics, University of Johannesburg, Johannesburg, South Africa. Introduction Siulation of physical systes often requires the solution of a syste of ordinary differential equations, in the for of an initial-value proble. Usually, a Runge-Kutta ethod is used to solve such a syste nuerically. Recently, we eained how the coputational efficiency of a Runge-Kutta ethod could be iproved through the echanis of the RKrGL algorith, in the contet of global error control via reintegration (Prentice, 2009). The RKrGL ethod for solving the d-diensional syste dy f ( y, ) y( 0) y0 a b d = = (.) is based on an eplicit Runge-Kutta ethod of order r (RKr), and -point Gauss-Legendre quadrature (GL). The ethod has a global error of order r +, which is the sae order as the local order of the underlying RKr ethod, provided that r and are chosen such that r + 2 (Prentice, 2008). Of course, any ethod designed for solving IVPs ust facilitate local error control. In this paper we describe an effective algorith for controlling the local relative error in RKrGL. 2. Terinology and relevant concepts In this section we describe terinology and concepts relevant to the paper, including a brief description of the RKrGL ethod. Note that, throughout this paper, overbar, as in v, indicates an d vector, and caret, as in M, denotes an d datri. 2. Eplicit Runge-Kutta ethods We denote an eplicit RK ethod for solving (.) by ( ) wi wi hf i i, yi + = + (2.) where hi i+ i is a stepsize, w i denotes the nuerical approiation to y ( i ), and F(, y ) is a function associated with the particular RK ethod (indeed, F(, y ) could be regarded as the function that defines the ethod).

3 476 Nuerical Siulations - Applications, Eaples and Theory 2.2 Local and global errors We define the global error in any nuerical solution at i by and, specifically, the RK local error at i by Δi wi yi (2.2) (, ) εi yi + hi F i yi yi In the above, yi and y i are the true solutions at i and i, respectively. Note that the true value yi is used in the bracketed ter in (2.3). y ' = f, y we have Note also that for the derivative ( ) (, ) (, ) (, ) (, ) (2.3) f i wi = f i yi +Δ i = f i yi + fy i ϑ Δ i i (2.4) ϑ in ( ) In the above we use the sybol i fy i, ϑi Δ i siply to denote an appropriate set of constants such that fy ( i, ϑi) Δ i is the residual ter in the first-order Taylor epansion f i, y i +Δ i. Furtherore, f y is the Jacobian of ( ) where { f f f } f f y y d f y = fd fd y y d, 2,, d are the coponents of f, and d is the diension of the syste (.). Clearly, a global error of Δ i in i O Δ in the derivative f ( i, w i). w iplies an error of ( ) i 2.3 Gauss-Legendre quadrature Gauss-Legendre quadrature on uv, with nodes is given by (Kincaid & Cheney, 2002) v (2.5) 2+ f (, y) d = h Cif ( i, yi ) + O( h ) (2.6) u where the nodes i are the roots of the Legendre polynoial of degree on uv,. Here, h is the average separation of the nodes on uv,, a notation we will adopt fro now on, and the C i are appropriate weights. The average node separation h on uv, is defined by v u h. + The nodes on [,], denoted i, are apped to corresponding nodes i on uv, via (2.7) i = ( v u ) i + u+ v, 2 (2.8)

4 A General Algorith for Local Error Control in the RKrGL Method 477 and the weights C i are constants on any interval of integration. We have referred to the interval [,] above because the nodes i on this interval are etensively tabulated. 2.4 The RKrGL algorith We briefly describe the general RKrGL algorith on the interval ab,, with reference to Figure. RK GL RK GL... a= 0 p p+... p+ 2p... b H H 2 Fig.. Scheatic depiction of the RKrGL algorith. Subdivide ab, into N subintervals H j. At the RK nodes on ( ) ( ) ( ) H j we use RKr: { } wi+ = wi + hf i i, wi i j p,, j p +. (2.9) At the GL nodes we use -point GL quadrature: where μ = 0,, 2,. Note that p +. The GL coponent is otivated by ( μ+ ) p μp w = w + h C f, w. (2.0) ( μ + ) p μp i ( i+ μp i+ μp) ( ) ( μ + ) p μp i ( i+ μp i+ μp) f, y d = y y h C f, y ( μ + ) p μp i ( i+ μp, i+ μp). y y + h C f y (2.) The RKrGL algorith has been shown to be consistent, convergent and zero-stable (Prentice, 2008). 2.5 Local error at the GL nodes The local error at the GL nodes is defined in a siilar way to that for an RK ethod: μp+ + = ( μ+ ) p μp ( ) 2+ ( ) ( ) f, y d = y y = h C f, y + O h μp+ + μp i i+ μp i+ μp ( μ+ ) p 2+ ε( μ+ ) p yμp + h Cif ( i+ μp, yi+ μp) y( μ+ ) p = O( h ). (2.2)

5 478 Nuerical Siulations - Applications, Eaples and Theory We reind the reader that in RKrGL we choose r and such that r + 2, which r O h + (Prentice, 2008). ensures that RKrGL has a global error of ( ) 2.6 Ipleentation of RKrGL There are a few points regarding the ipleentation of RKrGL that need to be discussed: a. If we erely saple the solutions at the GL nodes, treating the coputations at the RK nodes as if they were the stages of an ordinary RK ethod, then RKrGL would be reduced to an inefficient one-step ethod. This is not the intention behind the developent of RKrGL; rather, RKrGL represents an attept to iprove the efficiency of any RKr ethod, siply by replacing the coputation at every ( + )th node by a quadrature forula which does not require evaluation of any of the stages in the underlying RKr ethod. b. Of course, it is clear fro the above that on H the RK nodes are required to be consistent with the nodes necessary for GL quadrature. If, however, the RK nodes are located differently (as would be required by a local error control echanis, for eaple) then it is a siple atter to construct a Herite interpolating polynoial of degree 2 + (which has an error of order ) using the solutions at the nodes { 0,, }. Then, assuing 0 aps to and aps to the largest Legendre polynoial root on [,], the position of the other nodes {,, } suitable for GL quadrature ay be deterined, and the Herite polynoial ay be used to find approiate solutions of order r + at these nodes, thus facilitating the GL coponent of RKrGL. A siilar procedure is carried out on the net subinterval H, and so on. Indeed, we will see that the Herite polynoial described here will play an iportant role in our error control process, and is described in ore detail in the net subsection. c. If the underlying RKr ethod possesses a continuous etension it would not be necessary to construct the Herite polynoial described above. However, there is no guarantee that a continuous etension of appropriate order (at least 2 + ) will be available, and it is generally true that deterining a continuous etension for a RK ethod requires additional stages in the RK ethod, which would ost likely coproise the gain in efficiency offered by RKrGL. Note that the construction of the Herite polynoial only requires one additional evaluation of f ( y, ), at. 2.7 The Herite interpolating polynoial If the data { i, yi, y i : i = 0,, } are available, then a polynoial HP( ), of degree at ost 2 +, with the interpolatory properties HP( i) = yi H P( i) = y i (2.3) for each i, ay be constructed. If the nodes i are distinct, then HP( ) is unique. This approiating polynoial is known as the Herite interpolating polynoial (Burden & Faires, 200) and has an approiation error given by + ( ξ ( ) ) ( 2 2 )! (2 2) y 2 y( ) HP( ) = ( i) 0 + (2.4)

6 A General Algorith for Local Error Control in the RKrGL Method 479 where 0 < ξ( ) <. If h is the average separation of the nodes on 0,, it is possible to write i = σ ih, where σ i is a suitable constant, and hence 2+ 2 y ( ) H ( ) = Oh ( ). (2.5) The algorith for deterining the coefficients of H P () is linear, as in P c= A b (2.6) where c is a vector of the coefficients of HP( ), A is the relevant interpolation atri, and b is a vector containing y i and y i. The details of these ters need not concern us here; rather, if an error O( Δ ) eists in each of y i and y i, then an error of O( Δ ) will eist in each coponent of c. Moreover, since HP( ) is linear in its coefficients, then an error of O( Δ ) will also eist in any coputed value of HP( ). Consequently, we ay write where the ( ) 2+ 2 y ( ) H ( ) = Oh ( ) + O( Δ ) (2.7) P O Δ ter arises fro errors in y i and y i. We have assued, of course, that the errors in y i and y i are of the sae order, which is the situation that we will encounter later. 3. Local error control in RKrGL 3. The order of the tande ethod The idea behind the use of a tande ethod is that it ust be of sufficiently high order such that, relative to the approiate solution generated by RKrGL, the tande ethod yields a solution that ay be assued to be essentially eact. This solution is propagated in both RKrGL and the tande ethod itself, and the difference between the two solutions is taken as an estiate of the local error in RKrGL. This aounts to so-called local etrapolation and is not dissiilar in spirit to error estiation techniques eployed using Runge-Kutta ebedded pairs (Hairer et al., 2000; Butcher, 2003). Generally speaking, though, the tande ethod is not ebedded. To decide on an appropriate order for the tande ethod we consider the local error at the GL nodes ε ( μ+ ) p = yμp + h Cif ( i+ μp, yi+ μp) y( μ+ ) p = ( wμpt, Δ μpt, ) + h Cif ( i+ μp, wi+ μpt, Δi+ μpt, ) w( μ+ ) p, t Δ( μ+ ), t where w ( is the solution fro the tande ethod at ),t ( ), and w. Epanding the ter in the su in a Taylor series gives ( ),t ε ( μ+ ) p = wμpt, + h Cif ( i+ μp, wi+ μpt, ) w( μ+ ) p, t Δ +Δ ( μ + ) p, t + h C f (, ζ ) Δ μp, t i y i+ μp i+ μp, t i+ μp, t ( p ) (3.) Δ ( is the global error in ),t (3.2)

7 480 Nuerical Siulations - Applications, Eaples and Theory and so wμpt, + h Cif ( i+ μp, wi+ μpt, ) w( μ + ) pt, μp, t i y i+ μp i+ μp, t i+ μp, t = ε ( μ+ ) p + Δ Δ( μ+ ) p, t h C f (, ζ ) Δ where ζ i + μp, t is analogous to ϑ i in (2.4). The su on the RHS of (3.3) is of higher order than Δ μpt, Δ ( μ+ ) p, t, because of the ultiplication by h, and since we cannot epect, in q,, 0, the ter in parentheses ust be O( h ) 2+ global order of the tande ethod. Since ε ( μ ) p O( h ) general, that Δμpt Δ ( μ+ ) pt= require q > 2+ in order for (3.3), where q is the + = in the RKrGL ethod, we ( +, + ) ( + ), ε( + ) w + h C f w w (3.4) to be a good (and asyptotically ( 0) μpt, i i μp i μpt, μ p t μ p h correct) estiate for the local error in RKrGL. The first two ters on the LHS of (3.4) arise fro RKrGL with the tande solution as input, while w ( μ + ) p, t is the tande solution at ( μ + ) p. The iplication, then, is that the tande ethod ust have a global order of at least 2 + 2, which iplies q > r + 2, since we already have r+ = 2 in RKrGL. We acknowledge that our choice of q differs fro conventional wisdo (which would choose q > r + so that the local order of the tande ethod is one greater than the RK local order), but it is clear fro (3.3) that the propagation of the tande solution requires the global order of the tande ethod to be greater than the order of ε ( μ + ) p. Of course, at the RK nodes the local order is r +, so the tande ethod with global order q > r + 2 is ore than suitable at these nodes. 3.2 The error control algorith We describe the error control algorith on the first subinterval H = [ 0( = a), p ] (see Figure ). The sae procedure is then repeated on subsequent subintervals. Solutions w,r and w,q are obtained at using RKr and RKq, respectively. We assue r+, r = 0, r, q w y L h w w (3.5) where h0 0 and L is a vector of local error coefficients (we will discuss the choice of a value for h 0 later). The eponent of r + indicates the order of the local error in RKr. We find the aiu value of w y, ri,, i y w, ri, w, qi, i =,, d (3.6) w, i, q, i

8 A General Algorith for Local Error Control in the RKrGL Method 48 where the inde i refers to the coponents of the indicated vectors (so w, ri, is the ith coponent of w,r, etc). Call this aiu M and say it occurs for i = k. Hence, w, rk, w, qk, M = (3.7) w, qk, is the largest relative error in the coponents of w,r. Note that k ay vary fro node to node, but at any particular node we will always intend for k to denote the aiu value of (3.6). We now deand that δ δ (3.8) M R w, r, k w, q, k R w, q, k where δ R is a user-defined relative tolerance. If this inequality is violated we find a new stepsize h such that 0 δ r,, R w + q k r+ 0 = 0.9, k( 0) < δ R, q, k L, k h L h w where L,k is the kth coponent of L, and then we find new solutions w,r and w,q using h 0 ( L is deterined fro (3.5)). The factor 0.9 in (3.9) is a safety factor allowing for the fact that w,q is not truly eact. To cater for the possibility that any coponent of w,q is close to zero we actually deand { δ δ }, rk,, qk, A R, qk, (3.9) w w a, w (3.0) where δ A is a user-defined absolute tolerance. We then set h = h0 and proceed to the node 2, where the error control process is repeated, and siilarly for 3 up to. The process of recalculating a solution using a new stepsize is known as a step rejection. In the event that the condition in (3.0) is satisfied, we still calculate a new stepsize h 0 (which would now be larger than h 0 ) and set h = h0, on the assuption that if h 0 satisfies (3.0) at, then it will do so at 2 as well (however, we also place an upper liit on h 0 of 2h 0, although the choice of the factor two here is soewhat arbitrary). In the worst-case scenario we would find that h is too large and a new, saller value h ust be used. The eception occurs when w, rk, w, qk, = 0. In this case we siply set h = 2h0 and proceed to 2. The above is nothing ore than well-known local relative error control in an eplicit RK ethod using local etrapolation. It is at the GL node p that the algorith deviates fro the nor. A step-by-step description of the procedure at p follows:. Once error control at {, 2,, } has been effected (which necessarily defines the positions of {, 2,, } due to stepsize odifications that ay have occurred), the location of p ust be deterined such that the local relative error at p is less than a δ, δ w, in which k has the eaning discussed earlier. { A R p, q, k }

9 482 Nuerical Siulations - Applications, Eaples and Theory 0 = corresponds to on the interval,, and corresponds to the largest root of the th-degree Legendre polynoial in,. This allows p ( = v) to be found, where p corresponds to on 2. To this end, we utilize the ap (2.8), deanding that ( u),, and so new nodes {, 2,, } { }, 2,,, are consistent with the GL quadrature nodes on 0, p. 3. We wish to perfor GL quadrature, using the nodes {, 2,,, } but we do not have the approiate solutions { w, q,, w, q} at {, 2,, } can be deterined such that 4. Hence, we construct the Herite interpolating polynoial P ( ) the original nodes { }, on 0, p,. H on 0, using, 2,, and the solutions that have been obtained at these nodes; of course, the derivative of y ( ) at these nodes is given by (, ) f y. Note that a Herite polynoial ust be constructed for each of the s coponents of the syste, so if d >, HP ( ) is actually a d vector of Herite polynoials. 5. We use the qth-order solutions that are available, so that we epect the approiation q error in each HP ( ) to be ( ) 6. The solutions { w, q,, w, q} at {, 2,, } { HP( ) ( )},, HP GL quadrature then gives w p with local error Oh ( ) O h, as shown in (2.4) and (2.7). are then obtained fro, as per (2.2). 8. The tande ethod RKq is used to find w p, q, and wp wpq, is then used for error control: a. we know that the local error in p separation on 0, p ; 2+ w is Oh ( ) b. if the local error is too large then a new average node separation c. if using p h, a new position for p, denoted, where h here is the average node p, is found fro >, we redefine the nodes {, 2,,, } these new nodes using H ( ) P, and then find solutions at h is deterined; p 0 = + ph ;, find qth-order solutions at p using GL quadrature and RKq; d. if p, we reject the GL step since there is now no point in finding a solution at p. 9. After all this, the node p or (if p ) defines the endpoint of the subinterval H ; the stepsize h is set equal to the largest separation of the nodes on H, and the

10 A General Algorith for Local Error Control in the RKrGL Method 483 entire error control procedure is ipleented on the net subinterval H 2. Note also that it is the qth-order solution at the endpoint of H that is propagated in the RK solution at the net node. 3.3 Initial stepsize To find a stepsize h 0 to begin the calculation process, we assue that the local error coefficient L, k = and then find h 0 fro ( { }) a δ, r A δ R y + k h0 0, = (3.) Solutions obtained with RKr and RKq using this stepsize then enable a new, possibly larger, h 0 to be deterined, and it is this new h 0 that is used to find the solutions w,r and w,q at the node. 3.4 Final node We keep track of the nodes that evolve fro the stepsize adjustents, until the end of the interval of integration b has been eceeded. We then backtrack to the node on ab, closest to b (call it f ), deterine the stepsize hf b f, and then find w br, and w bq,, the nuerical solutions at b using RKr and RKq, with h f, f and w f, q as input for both RKr and RKq. This copletes the error control procedure. 4. Coents on ebedded RK ethods and continuous etensions Our intention has been to develop an effective local error control algorith for RKrGL, and we believe that the above-entioned algorith achieves this objective. Moreover, the algorith is general in the choice of RKr and RKq. These two ethods could be entirely independent of each other, or they could constitute an ebedded pair, as in RK(r,q). This latter choice would require fewer stage evaluations at each RK node, and so would be ore efficient than if RKr and RKq were independent. Nevertheless, the use of an ebedded pair is not necessary for the proper functioning of our error control algorith. The option of constructing HP( ) using the nodes { = p, p,, 2p } for error control at 2p (as opposed to using { p,, 2p } ) is worth considering. Such a polynoial, together with the Herite polynoial constructed on { 0,,, }, fors a piecewise continuous approiation to y ( ) on [ 0, 2p ]. Of course, this process is repeated at the nodes { 2p, 2p+,, 3p }, and so on. In this way the Herite polynoials, which ust be constructed out of necessity for error control purposes, becoe a piecewise continuous (and sooth) etension of the approiate discrete solution. Such an etension is not constructed a posteriori; rather, it is constructed on each subinterval H i as the RKrGL algorith proceeds, and so ay be used for event trapping. 5. Nuerical eaples We will use RK5GL3 to deonstrate the error control algorith. In RK5GL3 we have r = 5, = 3 so that the tande ethod ust be an eighth-order RK ethod, which we

11 484 Nuerical Siulations - Applications, Eaples and Theory denote RK8. The RK5 ethod in RK5GL3 is due to Fehlberg (Hairer et al., 2000), as is RK8 (Hairer et al., 2000; Butcher, 2003). By way of eaple, we solve on 0,5 with ( ) y 0 = 0, and 2 y' = 2y 2 + (5.) y y y' = 4 20 on 0, 30 y 0 =. The first of these has a uniodal solution on the indicated interval, and we will refer to it as IVP. The second proble is one of the test probles used by Hull et al (Hull et al., 972), and we will refer to it as IVP2. These probles have solutions with ( ) (5.2) IVP: IVP2: y( ) = 2 + (5.3) 20 y( ) = + 9 /4 e In Table we show the results of ipleenting our local error control algorith in solving 0 both test probles. The absolute tolerance δ A was always 0, ecept for IVP with 0 2 = 0, for which = 0 was used. δ R δ A IVP δ R RK step rejections GL step rejections nodes RKGL subintervals IVP2 δ R RK step rejections GL step rejections nodes RKGL subintervals Table. Perforance data for error control algorith applied to IVP and IVP2. In this table, RK step rejections is the nuber of ties a saller stepsize had to be deterined at the RK nodes; GL step rejections is the nuber of ties that 4 3, as described in the previous section; nodes is the total nuber of nodes used on the interval of integration, including the initial node 0 ; and RKGL subintervals is the total nuber of subintervals H i used on the interval of integration. It is clear that as δ R is decreased so the nuber of nodes and RKGL subintervals increases (consistent with a decreasing stepsize), and so there is

12 A General Algorith for Local Error Control in the RKrGL Method 485 ore chance of step rejections. There are not any RK step rejections for either proble. 0 When δ R = 0 the GL step rejections for IVP are 9 out a possible 25 (alost 80%), but for IVP2 the GL step rejections nuber only about 38%). In both cases the GL step rejections arise as a result of relatively large local error coefficients at the GL nodes, which necessarily lead to relatively sall values of h, the average node separation, so that the situation 4 3 is quite likely to occur. Figures 2 and 3 show the RK5GL3 local error for IVP and IVP2. The curve labelled tolerance in each figure is δ R yi, which is the upper liit placed on the local error. 0-6 to leran ce IV P 0-7 Error 0-8 actual error Fig. 2. RKGL local error for IVP, with 6 δ R = to leran ce Error actual error IV P Fig. 3. RKGL local error for IVP2, with 8 δ R = In Figure 2 we have used δ R = 0, and in Figure 3 we have used δ R = 0. It is clear that in both cases the tolerance has been satisfied, and the error control algorith has been successful. In Figure 4, for interest's sake, we show the stepsize variation as function of node inde (#) for these two probles.

13 486 Nuerical Siulations - Applications, Eaples and Theory.2.0 IV P2 0.8 h IVP # Fig. 4. Stepsize h vs node inde (#) for IVP and IVP2. To deonstrate error control in a syste, we use RK5GL3 to solve y = y2 2 y 2 = e sin 2y + 2y2 2 3 y( 0 ) =, y2( 0) = 5 5 on 0,3. The solution to this syste, denoted SYS, is (5.4) 2 e y = ( sin 2 cos ) 5 2 e y2 = + 5 ( 4sin 3cos ) The perforance table for RK5GL3 local error control applied to this proble is shown in Table 2. SYS δ R RK step rejections GL step rejections nodes RKGL subintervals Table 2. Perforance data for error control algorith applied to SYS. The perforance is siilar to that shown in Table. In all calculations reflected in Table 2, 2 we have used δ A = 0. The error in the coponents y and y 2 of SYS is shown in Figures 5 and 6. (5.5)

14 A General Algorith for Local Error Control in the RKrGL Method actual error tolerance 0-7 Error SYS co ponent y Fig. 5. Error in coponent y of SYS actual error to leran ce 0-7 Error SYS co ponent y Fig. 6. Error in coponent y 2 of SYS. 7. Conclusion and scope for further research We have developed an effective algorith for controlling the local relative error in RKrGL, with r +. The algorith utilizes a tande RK ethod of order r + 3, at least. A few nuerical eaples have deonstrated the effectiveness of the error control procedure. 7. Further research Although the algorith is effective, it is soewhat inefficient, as evidenced by the large nuber of step rejections shown in the tables. Ways to iprove efficiency ight include : a. The use of an ebedded RK pair, such as DOPRI853 (Dorand & Prince, 980), to reduce the total nuber of RK stage evaluations, b. Using a high order RKGL ethod as the tande ethod, since the RKGL ethods were originally designed to iprove RK efficiency, c. Error control per subinterval H j, rather than per node, which ight require reintegration on each subinterval,

15 488 Nuerical Siulations - Applications, Eaples and Theory d. Optial stepsize adjustent, so that stepsizes that are saller than necessary are not used. Saller stepsizes iplies ore nodes, which iplies greater coputational effort. 8. References Burden, R.L. and Faires, J.D., (200), Nuerical analysis, 7th ed., Brooks/Cole, , Pacific Grove. Butcher, J.C., (2003), Nuerical ethods for ordinary differential equations, Wiley, , Chichester. Dorand, J.R. and Prince, P.J., A faily of ebedded Runge-Kutta forulae, Journal of Coputational and Applied Matheatics, 6 (980) 9-26, Hairer, E., Norsett, S.P. and Wanner, G., (2000), Solving ordinary differential equations I: Nonstiff probles, Springer-Verlag, , Berlin. Hull, T.E., Enright, W.H., Fellen, B.M. and Sedgwick, A.E., Coparing nuerical ethods for ordinary differential equations, SIAM Journal of Nuerical Analysis, 9, 4 (972) , Kincaid, D. and Cheney, W., (2002), Nuerical Analysis: Matheatics of Scientific Coputing, 3rd ed., Brooks/Cole, , Pacific Grove. Prentice, J.S.C., The RKGL ethod for the nuerical solution of initial-value probles, Journal of Coputational and Applied Matheatics, 23, 2 (2008) , Prentice, J.S.C.,Iproving the efficiency of Runge-Kutta reintegration by eans of the RKGL algorith, (2009), In: Advanced Technologies, Kankesu Jayanthakuaran, (Ed.), , INTECH, , Vukovar.

16 Nuerical Siulations - Applications, Eaples and Theory Edited by Prof. Lutz Angerann ISBN Hard cover, 520 pages Publisher InTech Published online 30, January, 20 Published in print edition January, 20 This book will interest researchers, scientists, engineers and graduate students in any disciplines, who ake use of atheatical odeling and coputer siulation. Although it represents only a sall saple of the research activity on nuerical siulations, the book will certainly serve as a valuable tool for researchers interested in getting involved in this ultidisciplinary ï eld. It will be useful to encourage further eperiental and theoretical researches in the above entioned areas of nuerical siulation. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following: Justin Prentice (20). A General Algorith for Local Error Control in the RKrGL Method, Nuerical Siulations - Applications, Eaples and Theory, Prof. Lutz Angerann (Ed.), ISBN: , InTech, Available fro: InTech Europe University Capus STeP Ri Slavka Krautzeka 83/A 5000 Rijeka, Croatia Phone: +385 (5) Fa: +385 (5) InTech China Unit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, , China Phone: Fa: