SUBROUTINE MATPDE( N, NX, NY, A, ROWPTR, COLIND, RHS ) * .. * .. Scalar Arguments .. INTEGER N, NX, NY * .. * .. Array Arguments .. INTEGER ROWPTR( * ), COLIND( * ) DOUBLE PRECISION A( * ), RHS( * ) * * Purpose * ======= * * Forms the five-point central differences operator for the elliptic * PDE * * -(P Ux)x -(Q Uy)y + R Ux + (R U)x + S Uy + (S U)y + T U = F * * where P, Q, R, S and T are the functions of x and y. The domain * is the unit square (0,1)x(0,1), and the boundary condition are * Dirichlet. * * The matrix is a block tridiagonal matrix, there are NY blocks of * size NX by NX on the diagonal (each block is a tridiagonal matrix), * and then NY-1 blocks of size NX by NX on the sub- and super-block * diagonal positions. * * Important note: the matrix A is stored in compressed ROW format. * * Arguments * ========= * * N (output) INTEGER * The order of matrix A. N = NX*NY * * NX (input) INTEGER * The number of mesh points in the X-axis * * NY (input) INTEGER * The number of mesh points in the Y-axis * * A (output) DOUBLE PRECISION array, dimension (NZ) * The numerical values of the nonzero elements of matrix. * where NZ is the number of nonzero elements in the generated * matrix. Specifically, NZ = NX*NY + (NX-1)*NY*2 + NX*(NY-1)*2. * * ROWPTR (output) INTEGER array, dimension (N+1) * The row start pointers. * * COLIND (output) INTEGER array, dimension (NZ) * the column indices. * * RHS (output) DOUBLE PRECISION array, dimension (N) * The right hand side b of the linear system Ax = b. * * * Internal parameters * =================== * * BETA, GAMMA are the coefficients of the elliptic operator, * BNDRY( 4 ) are the boundary Dirichlet conditions. These parameters * may be changed, which will result in different matrices and spectral * distribution. * * ================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, HALF PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 ) DOUBLE PRECISION BETA, GAMMA PARAMETER ( BETA = 2.0D+1, GAMMA = 0.0D+0 ) * .. * .. Scalar variables .. INTEGER I, J, IX, JY, INDEX, JD, IAI IMPLICIT REAL*8 (A-H,O-Z) * * .. Array variables .. DOUBLE PRECISION BNDRY( 4 ) * .. * .. External functions .. DOUBLE PRECISION F, PC, QC, RC, SC, U EXTERNAL F, PC, QC, RC, SC, U * .. * .. Intrinsic functions .. INTRINSIC MOD, DBLE * * set up initial boundary conditions * BNDRY( 1 ) = ZERO BNDRY( 2 ) = ZERO BNDRY( 3 ) = ZERO BNDRY( 4 ) = ZERO * * set up other parameters * N = NX*NY HX = ONE / (DBLE(NX) + ONE) HY = ONE / (DBLE(NY) + ONE) HY2 = HY*HY RA = (HY/HX)**2 RB = HY2/HX * * FORM ROWPTR. * ROWPTR(1) = 1 DO 10 I = 1,N IAI = 1 IF (I.GT.NX) \$ IAI = IAI+1 IF(MOD(I-1,NX).NE.0) \$ IAI = IAI+1 IF(MOD(I,NX).NE.0) \$ IAI = IAI+1 IF(I.LE.N-NX) \$ IAI = IAI+1 ROWPTR(I+1) = ROWPTR(I) + IAI 10 CONTINUE * J = 1 DO 110 JY=1,NY YJ = HY*DBLE(JY) DO 100 IX=1,NX * XI = HX*DBLE(IX) INDEX = IX + (JY-1)*NX RHS(INDEX) = HY2*F(BETA,GAMMA,XI,YJ) P12 = PC (XI + HALF*HX,YJ) PM12 = PC (XI - HALF*HX,YJ) Q12 = QC (XI,YJ + HALF*HY) QM12 = QC (XI,YJ - HALF*HY) RIJ = RC (BETA,XI,YJ) R1 = RC (BETA,XI + HX,YJ) RM1 = RC (BETA,XI - HX,YJ) SIJ = SC (GAMMA,XI,YJ) S1 = SC (GAMMA,XI,YJ + HY) SM1 = SC (GAMMA,XI,YJ - HY) * * DIAGONAL. * A(J) = RA*(P12 + PM12) + Q12 + QM12 + HY2*TC(XI,YJ) COLIND(J) = INDEX JD = J J = J+1 * * LOWEST BAND. * IF (JY.EQ.1) \$ GOTO 20 A(J) = - ( QM12 + HALF*HY*(SIJ+SM1) ) COLIND(J) = INDEX - NX J = J+1 * * SUB-DIAGONAL. * 20 IF (IX.EQ.1) \$ GOTO 30 A(J) = - ( RA*PM12 + HALF*RB*(RIJ+RM1) ) COLIND(J) = INDEX - 1 J = J+1 * * SUPER-DIAGONAL. * 30 IF (IX.EQ.NX) \$ GOTO 40 A(J) = -RA*P12 + HALF*RB*(RIJ+R1) COLIND(J) = INDEX + 1 J = J+1 * * HIGHEST BAND. * 40 IF (JY.EQ.NY) \$ GOTO 50 A(J) = -Q12 + HALF*HY*(SIJ+S1) COLIND(J) = INDEX + NX J = J+1 * 50 CONTINUE * * BOUNDARY CONDITIONS. * IF (JY.NE.1) \$ GOTO 60 COEF = - ( QM12 + HALF*HY*(SIJ+SM1) ) IF (BNDRY(1).EQ.0) \$ RHS(INDEX) = RHS(INDEX) - COEF*U(XI,YJ-HY) IF (BNDRY(1).EQ.1) \$ A(JD) = A(JD) + COEF * 60 IF (IX.NE.1) GOTO 70 COEF = - ( RA*PM12 + HALF*RB*(RIJ+RM1) ) IF (BNDRY(2).EQ.0) \$ RHS(INDEX) = RHS(INDEX) - COEF*U(XI-HX,YJ) IF (BNDRY(2).EQ.1) \$ A(JD) = A(JD) + COEF * 70 IF (IX.NE.NX) GOTO 80 COEF = -RA*P12 + HALF*RB*(RIJ+R1) IF (BNDRY(3).EQ.0) \$ RHS(INDEX) = RHS(INDEX) - COEF*U(XI+HX,YJ) IF (BNDRY(3).EQ.1) \$ A(JD) = A(JD) + COEF * 80 IF (JY.NE.NY) \$ GOTO 100 COEF = -Q12 + HALF*HY*(SIJ+S1) IF (BNDRY(4).EQ.0) \$ RHS(INDEX) = RHS(INDEX) - COEF*U(XI,YJ+HY) IF (BNDRY(4).EQ.1) \$ A(JD) = A(JD) + COEF * 100 CONTINUE * 110 CONTINUE * RETURN * * End of MATPDE * END * DOUBLE PRECISION FUNCTION PC (X,Y) DOUBLE PRECISION X, Y INTRINSIC EXP PC = EXP(-X*Y) RETURN END * DOUBLE PRECISION FUNCTION QC (X,Y) DOUBLE PRECISION X, Y INTRINSIC EXP QC = EXP(X*Y) RETURN END * DOUBLE PRECISION FUNCTION TC (X,Y) DOUBLE PRECISION X, Y DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) TC = ONE / (ONE + X + Y) RETURN END * DOUBLE PRECISION FUNCTION U (X,Y) DOUBLE PRECISION X, Y DOUBLE PRECISION PI PARAMETER (PI = 3.14159265358979D+0) INTRINSIC SIN, EXP U = X * EXP(X*Y) * SIN(PI*X) * SIN(PI*Y) RETURN END * DOUBLE PRECISION FUNCTION RC (BETA,X,Y) DOUBLE PRECISION BETA, X, Y RC = BETA * (X+Y) RETURN END * DOUBLE PRECISION FUNCTION SC (GAMMA,X,Y) DOUBLE PRECISION GAMMA, X, Y SC = GAMMA * (X+Y) RETURN END * DOUBLE PRECISION FUNCTION F (BETA, GAMMA, X,Y) DOUBLE PRECISION BETA, GAMMA, X, Y * * DOUBLE PRECISION PI PARAMETER (PI = 3.14159265358979D+0) DOUBLE PRECISION ONE, TWO PARAMETER (ONE = 1.0D+0, TWO = 2.0D+0) * DOUBLE PRECISION PXY, PXXY, QXY, QYXY, CXY, RXY, RXXY, \$ SXY, SYXY, EXY, SX, CX, SY, CY, A, AX, \$ AXX, AY, AYY * DOUBLE PRECISION PC, QC, RC, SC EXTERNAL PC, QC, RC, SC INTRINSIC COS, SIN, EXP * PXY = PC(X,Y) PXXY = -Y * EXP(-X*Y) QXY = QC(X,Y) QYXY = X * EXP(X*Y) CXY = ONE / ( ONE + X + Y) RXY = RC(BETA,X,Y) RXXY = BETA SXY = SC(GAMMA,X,Y) SYXY = GAMMA * EXY = EXP(X*Y) SX = SIN(PI*X) CX = COS(PI*X) SY = SIN(PI*Y) CY = COS(PI*Y) * A = X * EXY * SX * SY AX = EXY * (PI * X * CX + (ONE + X * Y) * SX) * SY AXX = EXY * (PI * (-PI * X * SX + (X * Y + TWO) * CX) + Y * SX) \$ * SY + Y * AX AY = EXY * (PI * CY + X * SY) * X * SX AYY = X * (EXY * PI * (-PI * SY + X * CY) * SX + AY) * F = - (PXY*AXX + QXY*AYY) + \$ (TWO*RXY - PXXY) * AX + (TWO*SXY - QYXY) * AY + * (RXXY + SYXY + CXY) * A * RETURN END