On A General Formula of Fourth Order Runge-Kutta Method

On A General Formula of Fourth Order Runge-Kutta Method Delin Tan Zheng Chen Abstract In this paper we obtain a general formula of Runge-Kutta method in order with a free parameter t By picking the value of t it can generate many RK methods in order including some known results Numerical calculation shows that those new formulas have the same accuracy as the classical RK has I Introduction and Main Results The Runge-Kutta algorithm is used for solving the numerical solution of the ordinary differential equation y ( x f ( x y with initial condition y( x0 w In 0 about 900 Runge and Kutta developed the following classical fourth order Runge-Kutta iterative method ( K K K K h ( K f ( xk K f ( xk h w k K h K f ( xk h w ( k K h K f x k h K h which has the accumulative error in order of O(h Since then many mathematicians have tried to develop more Runge-Kutta like methods in a variety of directions In 99 R England [] developed another fourth order Runge-Kutta method (See the book of L F Shampine R C Allen and S Pruess [] ( K K K h ( K f ( xk K f ( xk h w k K h K f ( xk h w ( k K h K h K f x k h K h K h Although the higher order of Runge-Kutta methods can increase the error accuracy but it would need to pay the cost of calculation complexity Therefore the most mathematicians think that the fourth order Runge-Kutta method is the most efficient method to solve numerical solution for IVP It is naturally to ask: Do there exist other fourth order Runge-Kutta methods? If yes then what are they look like? In this paper we obtain a general formula fourth order Runge- Kutta method with a free parameter as the following Journal of Mathematical Sciences & Mathematics Education Vol 7 No

K ( t K tk Kh K f ( xk ( K f ( xk h w k K h K f ( xk h w k ( Kh Kh t t t t K f ( xk h ( Kh K h where t is a free parameter Remark : Pick t then ( becomes the classical Runge-Kutta method ( Remark : Pick t then ( becomes England s Runge-Kutta method ( Remark : Pick t we get a new formula for Runge-Kutta method ( K K K K h K f ( xk ( K f ( xk h w k K h ( k k ( k k K f x h w Kh Kh K f x h w Kh Kh Remark : Pick t we get the second new formula for Runge-Kutta method ( K K K K h K f ( xk (5 K f ( xk h w k K h K f ( xk h w k K h K h K f ( xk h K h K h Remark 5: Pick t 5 we get the third new formula for Runge-Kutta method ( K K 5 K K h K f ( xk ( K f ( xk h w k K h K f ( xk h w k K 0 h K 5 h 5 K f ( xk h K h K h Journal of Mathematical Sciences & Mathematics Education Vol 7 No

II Numerical Computing Tests With the same step value of h we use the different fourth order Runge-Kutta methods including the classical one to calculate the numerical solution for several differential equations with initial value condition at x 0 Then comparing to the real analytic solutions to estimate the relative error at x Our first example is to calculate the numerical solution for y y x (0 x with y(0 and h 0 Comparing to the analytic solution y e x x x at x we obtain the following results: RK methods Classical R England Formula ( Formula (5 Formula ( Relative errors at x 0009% 0009% 0009% 0009% 0009% Our second example is to calculate the numerical solution for y xy x (0 x with y(0 and h 0 / x Comparing to the analytic solution y e x we obtain the following results: RK methods Classical R England Formula ( Formula (5 Formula ( Relative errors at x 0005% 0005% 0005% 0005% 0005% Our last example is to calculate the numerical solution for the second order differential equation y y sin x (0 x with y(0 y (0 0 and h 0 y Define vector variabley y and apply the different fourth order Runge- Kutta methods to the following vector differential equation dy f ( x Y with initial condition Y (0 dx 0 0 0 where f ( x Y Y 0 sin x x After comparing to the analytic solution y e sin x we get the following results RK methods Classical R England Formula ( Formula (5 Formula ( Relative errors at x 0005% 0005% 0005% 0005% 0005% The above calculating results show that the new fourth order Runge-Kutta methods formulas ( (5 and ( have the exactly same error accuracy as the classical method ( does Journal of Mathematical Sciences & Mathematics Education Vol 7 No

III Lemma To derive the above general formula ( we need the following lemma: Lemma: Suppose that a b c d p p p q q a q b q a and q b are real numbers and function f(x y is continues in a neighborhood of a point (x 0 y 0 Let y y(x be a solution of differential equation y ( x f ( x y with initial condition y( x0 y Define 0 K f ( x y (7 K f ( x ph y qkh K f ( x ph y qakh qbkh K f ( x ph y qak h qbkh For small h > 0 if 5 (8 y( x h y( x ( ak bk ck dk h O( h is true for any continuous function f(x y in the domain of (x y then a b c d p q p qa qa p qa qb (9 ( bp cp dp ( bp cp dp ( bp cp dp ( cpq b dpq a dpqb dpq bqb ( p p dp pqa dp pqb dp p p 5 Proof: First pick y(x x then f(x y y'(x and K K K K Plug them into (8 we get 5 x h x ( a b c d h O( h Therefore we obtain ( a b c d y Still consider but setup f ( x y y ( x Then x y y qkh x qh K K x K x q K h q K h x q K h q K h K a b a b x ph x ph To make the point ( xk on the solution line y x K should be equal one Therefore the only possible relation between p and q is ( p q Journal of Mathematical Sciences & Mathematics Education Vol 7 No

x q q h K x ph For the same arguments on K we obtain ( p qa q b Then ( a b ( a b Then x q q h K x ph Similarly we obtain ( p qa q b If pick y(x x and setup f(x y y'(x x then K x K ( x ph K ( x ph K ( x ph Plug into (8 5 ( x h x h ax b( c( d( O( h [ ] ( 5 x ( a b c d xh bp cp dp h O( h Then we get bp cp dp (5 ( Next pick y(x x and setup f(x y y'(x x then K x K ( x ph K ( x ph K ( x ph Plug into (8 5 ( x h x h ax b( x ph c( x ph d( x ph O( h Then we get ( 5 x x h xh bp cp dp h O( h ( ( bp cp dp If pick y(x x and setup f(x y y'(x x then K x K ( x ph K ( x ph K ( x ph Plug into (8 5 ( x h x h ax b( x ph c( x ph d( x ph O( h Then we get ( 5 x x h x h xh bp cp dp h O( h (7 ( bp cp dp Again pick y( x x but setup y f ( x y y ( x Then K x and x Journal of Mathematical Sciences & Mathematics Education Vol 7 No 5

K ( y q K h ( x px h x p h x p h ( x pxh ph O( h ( y qakh qbkh K K ( x q ax h q bx h p q bxh O( h ( x p x h p q bxh O( h ( x pxh ph pq bh O( h ( y qakh qbkh ( x q ax h p q axh q bx h p q bxh O( h ( x px h pq a pqb xh O( h ( ( x pxh x ph pq a pqb h O( h Plug into (8 ( x h x ax h bx pxh ( ph h ( cx pxh ph pq bh h ( ( dx pxh x ph pq a pqb h h O( h ( x ( a b c d x h bp cp dp xh ( ( a b b Then we get ( bp cp dp ( dpq a dpqb cpq b From (7 8 dp q dp q cp q bp cp dp Therefore bp cp dp dp q dp q cp q h O( h ( a b b ( (8 ( cp q dp q dp q b a b Journal of Mathematical Sciences & Mathematics Education Vol 7 No

Still consider K y( x ( y qkh K x and setup ( y q K h q K h a b y f ( x y y ( x then K x ( ( x qax h qb x px h ph x h O h x px h pq bx h p qbxh O( h ( ( ( x p h x p h x p h ( pq bxh p pqbh p qbh O( h Because y q K h q K h Then K a b ( ( x q a x ph x ph xh O( h ( ( q b x ph x ph x pq bxh h O( h x q x h p q x h p q xh q x h a a p q x h p q xh 8 p q q xh O( h b b b b x p x h p q p q a ( b ( a b 8 pq bqb xh O( h ( y q K h q K h a b x h p q p q xh ( ( x and x px h pq a pq b x h p qa pqb xh 8pq bqb xh O( h ( ( ( ( ( ( x ph x ph x ph pq a pq b xh pq a pqb ph Plug into (8 p qa pq b h 8pq bqbh O( h Journal of Mathematical Sciences & Mathematics Education Vol 7 No 7

( ( ( ( ( ( x h x ax h bx px h ph x ph h cx ph x ph x ph pq bxh p pqbh p qbh h ( ( ( ( dx ph x ph x ph pq a pqb xh h ( a b ( a b 5 d pq pq ph p q pq h 8pq bq bh h O( h ( x a b c d x h ( bp cp dp x h ( ( b a b bp cp dp cp q dp q dp q xh bp cp dp h ( b a b ( b a b [ 5 dp pq a dp pqb dp p p 8 dp pqb ] h O( h ( [ dp p q dp p q dp p p dp q q ] h 5 O h p cp q dp q dp q p cp q dp q dp q h x x h x h xh h h p p h 8 ( a b b b Therefore p p dp p q dp p q dp p q 8dp q q ( a a b b b We obtain (8 p p dp p q dp p q dp p p dp q q ( a b b b Thus we have obtained all equations in (9 of the lemma Remark : Equations in lemma are the necessary conditions to make the RK constants a b c d p p p q q a q b q a and q b They might not be enough make (8 to be true for some f(xy Remark : Equations in (9 plus the following equations dpq bqb ( ( cpq b dpq a dpqb 8( cp pqb dp pqa dp pqb would be enough to make (8 to be true for all f(xy See the book of J H Mathews and K D Fink [] IV Derivation the General RK formulas The total number of questions in systems (9 and ( is The number of all unknown constants is also Therefore it could have only one solution for constants a b c d p p p q q a q b q a and q b theoretically How if we set up p p or p p or p p then the rank of the following equation system Journal of Mathematical Sciences & Mathematics Education Vol 7 No 8

( bp cp dp ( bp cp dp ( bp cp dp Would be reduced to two then we would get a free parameter By a tedious analyzing we have found that the only possible case is p p The further calculation reveals that this common value is only possibly to be / or / If p p / we can calculate to get p If p p / we can calculate to get p 5/ However the second choice p p / p 5/ is contradiction to equation system ( We omit the detail calculation here Now we choose p p and p to get b c d cp q dp q dp q becomes b c b c d a d b c d Then the equation ( b a b ( cq b qa qb Because q q p we get a b ( cq b cq b Set q b and t as a free parameter We obtain t t c and b c ( t From the equation we get Therefore ( p p dp p q dp p q dp p p dp q q a b b b ( q a q b q t b 5 ( b q t q t t b and q a Then we obtain the general RK formula with a parameter t Journal of Mathematical Sciences & Mathematics Education Vol 7 No 9

K ( t K tk Kh K f ( xk K f ( xk h w k K h K f ( xk h w k ( Kh Kh t t t t K f ( xk h ( Kh K h Moreover we can check that all constants in this formula satisfy conditions of ( for any constant t That proves that is a general Runge-Kutta method in order of Actually there is no other Runge-Kutta method in order of under the formation (7 Delin Tan PhD Southern University at New Orleans USA Zheng Chen PhD Southern University at New Orleans USA References [] R England Error estimate for Runge-Kutta type solutions of ordinary differential equation Computer Journal (99-70 [] L F Shampine R C Allen and S Pruess Fundamentals of Numerical Computing (997 John Wiley & Sons [] J H Mathews and K D Fink Numerical Methods: Using Matlab Fourth Edition (00 Prentice-Hall Pub Inc Journal of Mathematical Sciences & Mathematics Education Vol 7 No 0