added higher-order Adams formula.

added up to 8th order Adams to oscillation test.
jacobwilliams · Dec 30, 2015 · 360c25a · 360c25a · szaghi · Dec 30, 2015
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diff --git a/src/lib/foodie_integrator_adams_bashforth.f90 b/src/lib/foodie_integrator_adams_bashforth.f90
@@ -1,7 +1,7 @@
-!< FOODIE integrator: provide an explicit class of Adams-Bashforth multi-step schemes, from 1st to 4rd order accurate.
+!< FOODIE integrator: provide an explicit class of Adams-Bashforth multi-step schemes, from 1st to 16th order accurate.
 module foodie_integrator_adams_bashforth
 !-----------------------------------------------------------------------------------------------------------------------------------
-!< FOODIE integrator: provide an explicit class of Adams-Bashforth multi-step schemes, from 1st to 4rd order accurate.
+!< FOODIE integrator: provide an explicit class of Adams-Bashforth multi-step schemes, from 1st to 16th order accurate.
 !<
 !< Considering the following ODE system:
 !<
@@ -54,6 +54,7 @@ module foodie_integrator_adams_bashforth
 !< +\frac{37}{24} R(t^{n+1}, U^{n+1}) - \frac{9}{24} R(t^{n}, U^{n}) \right] $$
 !<
 !<#### Bibliography
+!<  * J. L. Maury Jr., G. P. Segal, "[Cowell Type Numerical Integration As Applied to Satellite Orbit Computation](http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690017325.pdf)", NASA-TM-X-63542, X-553-69-46, April 1969.
 !<
 !-----------------------------------------------------------------------------------------------------------------------------------
 
@@ -70,7 +71,7 @@ module foodie_integrator_adams_bashforth
 
 !-----------------------------------------------------------------------------------------------------------------------------------
 type :: adams_bashforth_integrator
-  !< FOODIE integrator: provide an explicit class of Adams-Bashforth multi-step schemes, from 1st to 4rd order accurate.
+  !< FOODIE integrator: provide an explicit class of Adams-Bashforth multi-step schemes, from 1st to 16th order accurate.
   !<
   !< @note The integrator must be created or initialized (initialize the *b* coefficients) before used.
   private
@@ -100,22 +101,157 @@ elemental subroutine init(self, steps)
   select case(steps)
   case(1)
     ! AB(1) Forward-Euler
-    self%b(1) = 1._R_P
+    self%b(1) = 1.0_R_P
   case(2)
-    ! AB(2)
-    self%b(1) = -0.5_R_P
-    self%b(2) = 1.5_R_P
+    self%b(1) = -1.0_R_P/2.0_R_P
+    self%b(2) = 3.0_R_P/2.0_R_P
   case(3)
-    ! AB(3)
-    self%b(1) =  5._R_P/12._R_P
-    self%b(2) = -4._R_P/3._R_P
-    self%b(3) = 23._R_P/12._R_P
+    self%b(1) = 5.0_R_P/12.0_R_P
+    self%b(2) = -16.0_R_P/12.0_R_P
+    self%b(3) = 23.0_R_P/12.0_R_P
   case(4)
-    ! AB(4)
-    self%b(1) =  -3._R_P/8._R_P
-    self%b(2) =  37._R_P/24._R_P
-    self%b(3) = -59._R_P/24._R_P
-    self%b(4) =  55._R_P/24._R_P
+    self%b(1) = -9.0_R_P/24.0_R_P
+    self%b(2) = 37.0_R_P/24.0_R_P
+    self%b(3) = -59.0_R_P/24.0_R_P
+    self%b(4) = 55.0_R_P/24.0_R_P
+  case(5)
+    self%b(1) = 251.0_R_P/720.0_R_P
+    self%b(2) = -1274.0_R_P/720.0_R_P
+    self%b(3) = 2616.0_R_P/720.0_R_P
+    self%b(4) = -2774.0_R_P/720.0_R_P
+    self%b(5) = 1901.0_R_P/720.0_R_P
+  case(6)
+    self%b(1) = -475.0_R_P/1440.0_R_P
+    self%b(2) = 2877.0_R_P/1440.0_R_P
+    self%b(3) = -7298.0_R_P/1440.0_R_P
+    self%b(4) = 9982.0_R_P/1440.0_R_P
+    self%b(5) = -7923.0_R_P/1440.0_R_P
+    self%b(6) = 4277.0_R_P/1440.0_R_P
+  case(7)
+    self%b(1) = 19087.0_R_P/60480.0_R_P
+    self%b(2) = -134472.0_R_P/60480.0_R_P
+    self%b(3) = 407139.0_R_P/60480.0_R_P
+    self%b(4) = -688256.0_R_P/60480.0_R_P
+    self%b(5) = 705549.0_R_P/60480.0_R_P
+    self%b(6) = -447288.0_R_P/60480.0_R_P
+    self%b(7) = 198721.0_R_P/60480.0_R_P
+  case(8)
+    self%b(1) = -36799.0_R_P/120960.0_R_P
+    self%b(2) = 295767.0_R_P/120960.0_R_P
+    self%b(3) = -1041723.0_R_P/120960.0_R_P
+    self%b(4) = 2102243.0_R_P/120960.0_R_P
+    self%b(5) = -2664477.0_R_P/120960.0_R_P
+    self%b(6) = 2183877.0_R_P/120960.0_R_P
+    self%b(7) = -1152169.0_R_P/120960.0_R_P
+    self%b(8) = 434241.0_R_P/120960.0_R_P
+  case(9)
+    self%b(1) = 1070017.0_R_P/3628800.0_R_P
+    self%b(2) = -9664106.0_R_P/3628800.0_R_P
+    self%b(3) = 38833486.0_R_P/3628800.0_R_P
+    self%b(4) = -91172642.0_R_P/3628800.0_R_P
+    self%b(5) = 137968480.0_R_P/3628800.0_R_P
+    self%b(6) = -139855262.0_R_P/3628800.0_R_P
+    self%b(7) = 95476786.0_R_P/3628800.0_R_P
+    self%b(8) = -43125206.0_R_P/3628800.0_R_P
+    self%b(9) = 14097247.0_R_P/3628800.0_R_P
+  case(10)
+    self%b(1) = -2082753.0_R_P/7257600.0_R_P
+    self%b(2) = 20884811.0_R_P/7257600.0_R_P
+    self%b(3) = -94307320.0_R_P/7257600.0_R_P
+    self%b(4) = 252618224.0_R_P/7257600.0_R_P
+    self%b(5) = -444772162.0_R_P/7257600.0_R_P
+    self%b(6) = 538363838.0_R_P/7257600.0_R_P
+    self%b(7) = -454661776.0_R_P/7257600.0_R_P
+    self%b(8) = 265932680.0_R_P/7257600.0_R_P
+    self%b(9) = -104995189.0_R_P/7257600.0_R_P
+    self%b(10) = 30277247.0_R_P/7257600.0_R_P
+  case(11)
+    self%b(1) = 134211265.0_R_P/479001600.0_R_P
+    self%b(2) = -1479574348.0_R_P/479001600.0_R_P
+    self%b(3) = 7417904451.0_R_P/479001600.0_R_P
+    self%b(4) = -22329634920.0_R_P/479001600.0_R_P
+    self%b(5) = 44857168434.0_R_P/479001600.0_R_P
+    self%b(6) = -63176201472.0_R_P/479001600.0_R_P
+    self%b(7) = 63716378958.0_R_P/479001600.0_R_P
+    self%b(8) = -46113029016.0_R_P/479001600.0_R_P
+    self%b(9) = 23591063805.0_R_P/479001600.0_R_P
+    self%b(10) = -8271795124.0_R_P/479001600.0_R_P
+    self%b(11) = 2132509567.0_R_P/479001600.0_R_P
+  case(12)
+    self%b(1) = -262747265.0_R_P/958003200.0_R_P
+    self%b(2) = 3158642445.0_R_P/958003200.0_R_P
+    self%b(3) = -17410248271.0_R_P/958003200.0_R_P
+    self%b(4) = 58189107627.0_R_P/958003200.0_R_P
+    self%b(5) = -131365867290.0_R_P/958003200.0_R_P
+    self%b(6) = 211103573298.0_R_P/958003200.0_R_P
+    self%b(7) = -247741639374.0_R_P/958003200.0_R_P
+    self%b(8) = 214139355366.0_R_P/958003200.0_R_P
+    self%b(9) = -135579356757.0_R_P/958003200.0_R_P
+    self%b(10) = 61633227185.0_R_P/958003200.0_R_P
+    self%b(11) = -19433810163.0_R_P/958003200.0_R_P
+    self%b(12) = 4527766399.0_R_P/958003200.0_R_P
+  case(13)
+    self%b(1) = 703604254357.0_R_P/2615348736000.0_R_P
+    self%b(2) = -9160551085734.0_R_P/2615348736000.0_R_P
+    self%b(3) = 55060974662412.0_R_P/2615348736000.0_R_P
+    self%b(4) = -202322913738370.0_R_P/2615348736000.0_R_P
+    self%b(5) = 507140369728425.0_R_P/2615348736000.0_R_P
+    self%b(6) = -915883387152444.0_R_P/2615348736000.0_R_P
+    self%b(7) = 1226443086129408.0_R_P/2615348736000.0_R_P
+    self%b(8) = -1233589244941764.0_R_P/2615348736000.0_R_P
+    self%b(9) = 932884546055895.0_R_P/2615348736000.0_R_P
+    self%b(10) = -524924579905150.0_R_P/2615348736000.0_R_P
+    self%b(11) = 214696591002612.0_R_P/2615348736000.0_R_P
+    self%b(12) = -61497552797274.0_R_P/2615348736000.0_R_P
+    self%b(13) = 13064406523627.0_R_P/2615348736000.0_R_P
+  case(14)
+    self%b(1) = -1382741929621.0_R_P/5230697472000.0_R_P
+    self%b(2) = 19382853593787.0_R_P/5230697472000.0_R_P
+    self%b(3) = -126174972681906.0_R_P/5230697472000.0_R_P
+    self%b(4) = 505586141196430.0_R_P/5230697472000.0_R_P
+    self%b(5) = -1393306307155755.0_R_P/5230697472000.0_R_P
+    self%b(6) = 2793869602879077.0_R_P/5230697472000.0_R_P
+    self%b(7) = -4204551925534524.0_R_P/5230697472000.0_R_P
+    self%b(8) = 4825671323488452.0_R_P/5230697472000.0_R_P
+    self%b(9) = -4246767353305755.0_R_P/5230697472000.0_R_P
+    self%b(10) = 2854429571790805.0_R_P/5230697472000.0_R_P
+    self%b(11) = -1445313351681906.0_R_P/5230697472000.0_R_P
+    self%b(12) = 537247052515662.0_R_P/5230697472000.0_R_P
+    self%b(13) = -140970750679621.0_R_P/5230697472000.0_R_P
+    self%b(14) = 27511554976875.0_R_P/5230697472000.0_R_P
+  case(15)
+    self%b(1) = 8164168737599.0_R_P/31384184832000.0_R_P
+    self%b(2) = -122594813904112.0_R_P/31384184832000.0_R_P
+    self%b(3) = 859236476684231.0_R_P/31384184832000.0_R_P
+    self%b(4) = -3728807256577472.0_R_P/31384184832000.0_R_P
+    self%b(5) = 11205849753515179.0_R_P/31384184832000.0_R_P
+    self%b(6) = -24704503655607728.0_R_P/31384184832000.0_R_P
+    self%b(7) = 41280216336284259.0_R_P/31384184832000.0_R_P
+    self%b(8) = -53246738660646912.0_R_P/31384184832000.0_R_P
+    self%b(9) = 53471026659940509.0_R_P/31384184832000.0_R_P
+    self%b(10) = -41825269932507728.0_R_P/31384184832000.0_R_P
+    self%b(11) = 25298910337081429.0_R_P/31384184832000.0_R_P
+    self%b(12) = -11643637530577472.0_R_P/31384184832000.0_R_P
+    self%b(13) = 3966421670215481.0_R_P/31384184832000.0_R_P
+    self%b(14) = -960122866404112.0_R_P/31384184832000.0_R_P
+    self%b(15) = 173233498598849.0_R_P/31384184832000.0_R_P
+  case(16)
+    self%b(1) = -16088129229375.0_R_P/62768369664000.0_R_P
+    self%b(2) = 257650275915823.0_R_P/62768369664000.0_R_P
+    self%b(3) = -1934443196892599.0_R_P/62768369664000.0_R_P
+    self%b(4) = 9038571752734087.0_R_P/62768369664000.0_R_P
+    self%b(5) = -29417910911251819.0_R_P/62768369664000.0_R_P
+    self%b(6) = 70724351582843483.0_R_P/62768369664000.0_R_P
+    self%b(7) = -129930094104237331.0_R_P/62768369664000.0_R_P
+    self%b(8) = 186087544263596643.0_R_P/62768369664000.0_R_P
+    self%b(9) = -210020588912321949.0_R_P/62768369664000.0_R_P
+    self%b(10) = 187463140112902893.0_R_P/62768369664000.0_R_P
+    self%b(11) = -131963191940828581.0_R_P/62768369664000.0_R_P
+    self%b(12) = 72558117072259733.0_R_P/62768369664000.0_R_P
+    self%b(13) = -30607373860520569.0_R_P/62768369664000.0_R_P
+    self%b(14) = 9622096909515337.0_R_P/62768369664000.0_R_P
+    self%b(15) = -2161567671248849.0_R_P/62768369664000.0_R_P
+    self%b(16) = 362555126427073.0_R_P/62768369664000.0_R_P
   endselect
   return
   !---------------------------------------------------------------------------------------------------------------------------------

diff --git a/src/lib/foodie_integrator_adams_bashforth_moulton.f90 b/src/lib/foodie_integrator_adams_bashforth_moulton.f90
@@ -1,8 +1,8 @@
-!< FOODIE integrator: provide a predictor-corrector class of Adams-Bashforth-Moutlon multi-step schemes, from 1st to 4rd order
+!< FOODIE integrator: provide a predictor-corrector class of Adams-Bashforth-Moutlon multi-step schemes, from 1st to 16th order
 !< accurate.
 module foodie_integrator_adams_bashforth_moulton
 !-----------------------------------------------------------------------------------------------------------------------------------
-!< FOODIE integrator: provide a predictor-corrector class of Adams-Bashforth-Moutlon multi-step schemes, from 1st to 4rd order
+!< FOODIE integrator: provide a predictor-corrector class of Adams-Bashforth-Moutlon multi-step schemes, from 1st to 16th order
 !< accurate.
 !<
 !< Considering the following ODE system:
@@ -92,7 +92,7 @@ module foodie_integrator_adams_bashforth_moulton
 
 !-----------------------------------------------------------------------------------------------------------------------------------
 type :: adams_bashforth_moulton_integrator
-  !< FOODIE integrator: provide an explicit class of Adams-Bashforth-Moulton multi-step schemes, from 1st to 4rd order accurate.
+  !< FOODIE integrator: provide an explicit class of Adams-Bashforth-Moulton multi-step schemes, from 1st to 16th order accurate.
   !<
   !< @note The integrator must be created or initialized (predictor and corrector schemes selection) before used.
   private