In recent times, the derivation of Runge-Kutta methods based on averages other than the arithme-tic mean is on the rise. In this paper, the authors propose a new version of explicit Runge-Kutta method, by introducing the harmonic mean as against the usual arithmetic averages in standard Runge-Kutta schemes.
During the last few decades, there has been a growing interest in problem solving systems based on the Runge- Kutta methods. Several methods have been developed using the idea different means such as the geometric mean, centroidal mean, harmonic mean, contra-harmonic mean and the heronian mean.
In previous papers [
The schemes introduced by [
where
and
where
Scheme (2) was referred to as RK-HM-AM. Using the definition of harmonic mean, the following scheme is proposed in this paper:
where,
where
The expansion of
Substituting (8), (9) and (10) into (4) and simplifying the resulting expression using MATHEMATICA (version 8.0.1) package, the coefficients of the powers of h in (4) are compared with that of the Taylors’ expansion of
Thus, the incremental function (4) of the proposed scheme is
and the proposed scheme (3) is
where
For the analysis of the absolute stability of the proposed 4sHERK scheme, the scalar test problem
Substituting (17)-(20) in (3) and simplifying the resulting expression results in,
Letting
The absolute stability region of the 4sHERK scheme is given in
Definition: The local truncation error at
And
Using the above definition together with (12), the local truncation error (LTE) of the proposed scheme is given as
where
Consider the IVP
with the theoretical solution
x | Exact Sol. | RK-4 | RK-HM-AM [ | RK-HM [ | 4sHERK | RK-4 Error | RK-HM-AM [ | RK-HM [ | 4sHERK Error |
---|---|---|---|---|---|---|---|---|---|
0.125 | 1.11803399 | 1.11803441 | 1.11803365 | 1.11803347 | 1.11803399 | 0.42308247e−6 | 0.3380746e−6 | 0.52268107e−6 | 0.37325369e−8 |
0.25 | 1.22474487 | 1.22474543 | 1.22474443 | 1.22474419 | 1.22474487 | 0.55362188e−6 | 0.44595043e−6 | 0.68606666e−6 | 0.44036872e−8 |
0.375 | 1.32287566 | 1.32287625 | 1.32287518 | 1.32287492 | 1.32287565 | 0.59022189e−6 | 0.47774995e−6 | 0.7328127e−6 | 0.44098678e−8 |
0.5 | 1.41421356 | 1.41421415 | 1.41421308 | 1.41421283 | 1.41421356 | 0.59242193e−6 | 0.48102403e−6 | 0.73644671e−6 | 0.42553943e−8 |
0.625 | 1.5 | 1.50000058 | 1.49999953 | 1.49999928 | 1.5 | 0.58129728e−6 | 0.47297202e−6 | 0.72321365e−6 | 0.4069489e−8 |
0.75 | 1.58113883 | 1.5811394 | 1.58113837 | 1.58113813 | 1.58113883 | 0.56516855e−6 | 0.46050983e−6 | 0.70355079e−6 | 0.38884294e−8 |
0.875 | 1.6583124 | 1.65831294 | 1.65831195 | 1.65831171 | 1.65831239 | 0.54755259e−6 | 0.44661313e−6 | 0.68190159e−6 | 0.37219154e−8 |
1.73205081 | 1.73205134 | 1.73205037 | 1.73205015 | 1.7320508 | 0.52998364e−6 | 0.43260686e−6 | 0.66022089e−6 | 0.35714376e−8 | |
0.125 | 1.80277564 | 1.80277615 | 1.80277522 | 1.802775 | 1.80277563 | 0.5131226e−6 | 0.41907842e−6 | 0.63936096e−6 | 0.34359544e−8 |
0.25 | 1.87082869 | 1.87082919 | 1.87082829 | 1.87082807 | 1.87082869 | 0.49722869e−6 | 0.4062709e−6 | 0.61966383e−6 | 0.33137697e−8 |
0.375 | 1.93649167 | 1.93649216 | 1.93649128 | 1.93649107 | 1.93649167 | 0.48237238e−6 | 0.39426268e−6 | 0.60123001e−6 | 0.32031635e−8 |
0.5 | 2.0 | 2.00000047 | 1.99999962 | 1.99999942 | 2.0 | 0.46853586e−6 | 0.38305324e−6 | 0.58404588e−6 | 0.31025875e−8 |
x | Exact Sol. | RK-4 | RK-HM-AM [ | RK-HM [ | 4sHERK | RK-4 Error | RK-HM-AM [ | RK-HM [ | 4sHERK Error |
---|---|---|---|---|---|---|---|---|---|
0.1 | 1.09544512 | 1.09544526 | 1.09544499 | 1.09544493 | 1.09544511 | 0.14972954e−6 | 0.12283314e−6 | 0.18686867e−6 | 0.89117402e−9 |
0.2 | 1.18321596 | 1.18321616 | 1.18321578 | 1.1832157 | 1.18321596 | 0.20809082e−6 | 0.17175178e−6 | 0.26033694e−6 | 0.1122773e−8 |
0.3 | 1.26491106 | 1.26491129 | 1.26491087 | 1.26491077 | 1.26491106 | 0.23071443e−6 | 0.19118572e−6 | 0.28910566e−6 | 0.11668382e−8 |
0.4 | 1.34164079 | 1.34164102 | 1.34164059 | 1.34164049 | 1.34164079 | 0.23787374e−6 | 0.19765556e−6 | 0.29840694e−6 | 0.11515036e−8 |
0.5 | 1.41421356 | 1.4142138 | 1.41421336 | 1.41421326 | 1.41421356 | 0.23792188e−6 | 0.19807497e−6 | 0.29870149e−6 | 0.11172532e−8 |
0.6 | 1.4832397 | 1.48323993 | 1.4832395 | 1.4832394 | 1.4832397 | 0.23461819e−6 | 0.19559589e−6 | 0.29472188e−6 | 0.10781778e−8 |
0.7 | 1.54919334 | 1.54919357 | 1.54919315 | 1.54919305 | 1.54919334 | 0.22976557e−6 | 0.19174751e−6 | 0.28874859e−6 | 0.10394097e−8 |
0.8 | 1.61245155 | 1.61245177 | 1.61245136 | 1.61245127 | 1.61245155 | 0.22426742e−6 | 0.18730473e−6 | 0.2819297e−6 | 0.10027719e−8 |
0.9 | 1.67332005 | 1.67332027 | 1.67331987 | 1.67331978 | 1.67332005 | 0.21858829e−6 | 0.18267092e−6 | 0.27485859e−6 | 0.96880082e−9 |
0.0 | 1.73205081 | 1.73205102 | 1.73205063 | 1.73205054 | 1.73205081 | 0.21296863e−6 | 0.17805796e−6 | 0.26784435e−6 | 0.9375225e−9 |
x | Exact Sol. | RK-4 | RK-HM-AM [ | RK-HM [ | 4sHERK | RK-4 Error | RK-HM-AM [ | RK-HM [ | 4sHERK Error |
---|---|---|---|---|---|---|---|---|---|
0.01 | 1.00995049 | 1.00995049 | 1.00995049 | 1.00995049 | 1.00995049 | 0.20121682e−11 | 0.18316459e−11 | 0.26254554e−11 | 0.2220446e−15 |
0.02 | 1.0198039 | 1.0198039 | 1.0198039 | 1.0198039 | 1.0198039 | 0.38344883e−11 | 0.34909853e−11 | 0.50035531e−11 | 0.44408921e−15 |
0.03 | 1.02956301 | 1.02956301 | 1.02956301 | 1.02956301 | 1.02956301 | 0.54869442e−11 | 0.49957816e−11 | 0.71600503e−11 | 0.44408921e−15 |
0.04 | 1.03923048 | 1.03923048 | 1.03923048 | 1.03923048 | 1.03923048 | 0.69868555e−11 | 0.63624661e−11 | 0.91180397e−11 | 0.66613381e−15 |
0.05 | 1.04880885 | 1.04880885 | 1.04880885 | 1.04880885 | 1.04880885 | 0.83497653e−11 | 0.76043616e−11 | 0.10897283e−10 | 0.66613381e−15 |
0.06 | 1.05830052 | 1.05830052 | 1.05830052 | 1.05830052 | 1.05830052 | 0.95894404e−11 | 0.87341245e−11 | 0.12515544e−10 | 0.66613381e−15 |
0.07 | 1.06770783 | 1.06770783 | 1.06770783 | 1.06770783 | 1.06770783 | 0.10717871e−10 | 0.97626351e−11 | 0.13988588e−10 | 0.66613381e−15 |
0.08 | 1.07703296 | 1.07703296 | 1.07703296 | 1.07703296 | 1.07703296 | 0.11745716e−10 | 0.10699663e−10 | 0.15330626e−10 | 0.66613381e−15 |
0.09 | 1.08627805 | 1.08627805 | 1.08627805 | 1.08627805 | 1.08627805 | 0.12682522e−10 | 0.11553869e−10 | 0.16553869e−10 | 0.66613381e−15 |
0.1 | 1.09544512 | 1.09544512 | 1.09544511 | 1.09544511 | 1.09544512 | 0.13536727e−10 | 0.12333246e−10 | 0.17669644e−10 | 0.66613381e−15 |
We apply the new 4sHERK method (13) to the above IVP and the results obtained are compared with the classical 4-stage fourth-order Runge-Kutta method and the methods of [
The results generated by the newly derived scheme in this paper evidently proved the extent of accuracy of the scheme in comparison with the other methods.
Evidently, the newly derived scheme is more accurate as seen from the computational results presented in