qAndLEvaluation Subroutine

private subroutine qAndLEvaluation(N_in, x, q, qder, L)

Evaluate the polynomial q=L_{N_in+1}-L_{N_in-1} and its derivative at position x in [-1,1] Recursive algorithm using the N_in-1 N_in-2 Legendre polynomials. (Algorithm 24, Kopriva book)

Arguments

Type IntentOptional Attributes Name
integer, intent(in) :: N_in

polynomial degree
real(kind=wp), intent(in) :: x

coordinate value in the interval [-1,1]
real(kind=wp), intent(out) :: q

\f$ q_N(\xi) \f$
real(kind=wp), intent(out) :: qder

\f$ \partial/\partial\xi \; L_N(\xi) \f$
real(kind=wp), intent(out) :: L

\f$ L_N(\xi) \f$

Called by

proc~~qandlevaluation~~CalledByGraph proc~qandlevaluation qAndLEvaluation proc~leggausslobnodesandweights LegGaussLobNodesAndWeights proc~leggausslobnodesandweights->proc~qandlevaluation

Source Code

SUBROUTINE qAndLEvaluation(N_in,x,q,qder,L)
IMPLICIT NONE
!----------------------------------------------------------------------------------------------------------------------------------
! INPUT/OUTPUT VARIABLES
INTEGER,INTENT(IN) :: N_in                            !! polynomial degree
REAL(wp),INTENT(IN)    :: x                               !! coordinate value in the interval [-1,1]
REAL(wp),INTENT(OUT)   :: L                               !! \f$ L_N(\xi) \f$
REAL(wp),INTENT(OUT)   :: q                               !! \f$ q_N(\xi) \f$
REAL(wp),INTENT(OUT)   :: qder                            !! \f$ \partial/\partial\xi \; L_N(\xi) \f$
!----------------------------------------------------------------------------------------------------------------------------------
! LOCAL VARIABLES
INTEGER            :: iLegendre
REAL(wp)               :: L_Nm1,L_Nm2                     ! L_{N_in-2},L_{N_in-1}
REAL(wp)               :: Lder,Lder_Nm1,Lder_Nm2          ! Lder_{N_in-2},Lder_{N_in-1}
!==================================================================================================================================
L_Nm2=1.0_wp
L_Nm1=x
Lder_Nm2=0.0_wp
Lder_Nm1=1.0_wp
DO iLegendre=2,N_in
  L=(REAL(2*iLegendre-1,wp)*x*L_Nm1 - REAL(iLegendre-1,wp)*L_Nm2)/REAL(iLegendre,wp)
  Lder=Lder_Nm2 + REAL(2*iLegendre-1,wp)*L_Nm1
  L_Nm2=L_Nm1
  L_Nm1=L
  Lder_Nm2=Lder_Nm1
  Lder_Nm1=Lder
END DO ! iLegendre
q=REAL(2*N_in+1,wp)/REAL(N_in+1,wp)*(x*L -L_Nm2) !L_{N_in+1}-L_{N_in-1} !L_Nm2 is L_Nm1, L_Nm1 was overwritten!
qder= REAL(2*N_in+1,wp)*L             !Lder_{N_in+1}-Lder_{N_in-1}
END SUBROUTINE qAndLEvaluation