/
first_derivative_sbp_operators.py
479 lines (318 loc) · 25.1 KB
/
first_derivative_sbp_operators.py
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def dx(ux, u, nx, dx, order):
# summation-by-parts finite difference operators for first derivatives du/dx
m = nx-1
# second order accurate case
if order==2:
# calculate partial derivatives on the boundaries:[0, m] with one-sided difference operators
ux[0, :] = (u[1, :] - u[0, :])/dx
ux[m, :] = (u[m, :] - u[m-1, :])/dx
#calculate partial derivatives in the interior:(1:nx-1)
for j in range(1, m):
ux[j, :] = (u[j+1, :] - u[j-1, :])/(2.0*dx)
# fourth order accurate case
if order==4:
#################################################
# calculate partial derivatives on the boundaries:(0,1,2,3, : m-3, m-2, m-1, m)
# with one-sided difference operators
ux[0,:] = -24./17*u[0,:] + 59./34*u[1, :] - 4./17*u[2, :] - 3./34*u[3,:]
ux[1,:] = -1./2*u[0,:] + 1./2*u[2, :] ;
ux[2,:] = 4./43*u[0,:] - 59./86*u[1, :] + 59./86*u[3, :] - 4./43*u[4,:]
ux[3,:] = 3./98*u[0,:] - 59./98*u[2, :] + 32./49*u[4, :] - 4./49*u[5,:]
ux[m,:] = 24./17*u[m,:] - 59./34*u[m-1, :] + 4./17*u[m-2, :] + 3./34*u[m-3,:]
ux[m-1,:] = 1./2*u[m,:] - 1./2*u[m-2, :] ;
ux[m-2,:] = -4./43*u[m,:] + 59./86*u[m-1, :]- 59./86*u[m-3, :]+ 4./43*u[m-4,:]
ux[m-3,:] = -3./98*u[m,:] + 59./98*u[m-2, :]- 32./49*u[m-4, :]+ 4./49*u[m-5,:]
#------------------------------------------------------------------------------------------------------------------------------
for i in range(4, m - 3):
ux[i,:] = 0.083333333333333*u[i-2,:] - 0.666666666666667*u[i-1,:] + 0.666666666666667*u[i+1,:] - 0.083333333333333*u[i+2,:]
ux[:,:] = ux/dx
# sixth order accurate case
#################################################
if order==6:
# calculate partial derivatives on the boundaries:(0,1,2,3,4,5,6 : m-6, m-5, m-4, m-3, m-2, m-1, m)
# with one-sided difference operators
ux[0,:] = -1.694834962162858*u[0,:] + 2.245634824947698*u[1,:] - 0.055649692295628*u[2,:] - 0.670383570370653*u[3,:]\
- 0.188774952148393*u[4,:] + 0.552135032829910*u[5,:] - 0.188126680800077*u[6,:]
ux[1,:] = -0.434411786832708*u[0,:] + 0.107043134706685*u[2,:] + 0.420172642668695*u[3,:] + 0.119957288069806*u[4,:]\
- 0.328691543801578*u[5,:] + 0.122487487014485*u[6,:] - 0.006557221825386*u[7,:]
ux[2,:] = 0.063307644169533*u[0,:] - 0.629491308812471*u[1,:] + 0.809935419586724*u[3,:] - 0.699016381364484*u[4,:]\
+ 0.850345731199969*u[5,:] - 0.509589652965290*u[6,:] + 0.114508548186019*u[7,:]
ux[3,:] = 0.110198643174386*u[0,:] - 0.357041083340051*u[1,:] - 0.117033418681039*u[2,:] + 0.120870009174558*u[4,:]\
+ 0.349168902725368*u[5,:] - 0.104924741749615*u[6,:] - 0.001238311303608*u[7,:]
ux[4,:] = 0.133544619364965*u[0,:] - 0.438678347579289*u[1,:] + 0.434686341173840*u[2,:] - 0.520172867814934*u[3,:]\
+ 0.049912002176267*u[5,:] + 0.504693510958978*u[6,:] - 0.163985258279827*u[7,:]
ux[5,:] = -0.127754693486067*u[0,:] + 0.393149407857401*u[1,:] - 0.172955234680916*u[2,:] - 0.491489487857764*u[3,:]\
- 0.016325050231672*u[4,:] + 0.428167552785852*u[6,:] - 0.025864364383975*u[7,:] + 0.013071869997141*u[8,:]
ux[6,:] = 0.060008241515128*u[0,:] - 0.201971348965594*u[1,:] + 0.142885356631256*u[2,:] + 0.203603636754774*u[3,:]\
- 0.227565385120003*u[4,:] - 0.590259111130048*u[5,:] + 0.757462553894374*u[7,:] - 0.162184436527372*u[8,:]\
+ 0.018020492947486*u[9,:]
ux[7,:] = 0.009910488565285*u[1,:] - 0.029429452176588*u[2,:] + 0.002202493355677*u[3,:] + 0.067773581604826*u[4,:]\
+ 0.032681945726690*u[5,:] - 0.694285851935105*u[6,:] + 0.743286642396343*u[8,:] - 0.148657328479269*u[9,:]\
+ 0.016517480942141*u[10,:]
ux[m-7,:] =-0.016517480942141*u[m-10,:] + 0.148657328479269*u[m-9,:] - 0.743286642396343*u[m-8,:] + 0.694285851935105*u[m-6,:]\
- 0.032681945726690*u[m-5,:] - 0.067773581604826*u[m-4,:] - 0.002202493355677*u[m-3,:] + 0.029429452176588*u[m-2,:]\
- 0.009910488565285*u[m-1,:]
ux[m-6,:] =-0.018020492947486*u[m-9,:] + 0.162184436527372*u[m-8,:] - 0.757462553894374*u[m-7,:] + 0.590259111130048*u[m-5,:]\
+ 0.227565385120003*u[m-4,:] - 0.203603636754774*u[m-3,:] - 0.142885356631256*u[m-2,:] + 0.201971348965594*u[m-1,:]\
- 0.060008241515128*u[m,:]
ux[m-5,:] =-0.013071869997141*u[m-8,:] + 0.025864364383975*u[m-7,:] - 0.428167552785852*u[m-6,:] + 0.016325050231672*u[m-4,:]\
+ 0.491489487857764*u[m-3,:] + 0.172955234680916*u[m-2,:] - 0.393149407857401*u[m-1,:] + 0.127754693486067*u[m,:]
ux[m-4,:] = 0.163985258279827*u[m-7,:] - 0.504693510958978*u[m-6,:] - 0.049912002176267*u[m-5,:] + 0.520172867814934*u[m-3,:]\
- 0.434686341173840*u[m-2,:] + 0.438678347579289*u[m-1,:] - 0.133544619364965*u[m,:]
ux[m-3,:] = 0.001238311303608*u[m-7,:] + 0.104924741749615*u[m-6,:] - 0.349168902725368*u[m-5,:] - 0.120870009174558*u[m-4,:]\
+ 0.117033418681039*u[m-2,:] + 0.357041083340051*u[m-1,:] - 0.110198643174386*u[m,:]
ux[m-2,:] =-0.114508548186019*u[m-7,:] + 0.509589652965290*u[m-6,:] - 0.850345731199969*u[m-5,:] + 0.699016381364484*u[m-4,:]\
- 0.809935419586724*u[m-3,:] + 0.629491308812471*u[m-1,:] - 0.063307644169533*u[m,:]
ux[m-1,:] = 0.006557221825386*u[m-7,:] - 0.122487487014485*u[m-6,:] + 0.328691543801578*u[m-5,:] - 0.119957288069806*u[m-4,:]\
- 0.420172642668695*u[m-3,:] - 0.107043134706685*u[m-2,:] + 0.434411786832708*u[m,:]
ux[m,:] = 0.188126680800077*u[m-6,:] - 0.552135032829910*u[m-5,:] + 0.188774952148393*u[m-4,:] + 0.670383570370653*u[m-3,:]\
+ 0.055649692295628*u[m-2,:] - 2.245634824947698*u[m-1,:] + 1.694834962162858*u[m,:]
for i in range(8, m-7):
ux[i,:] = -0.016666666666667*u[i-3,:] + 0.15*u[i-2,:] - 0.75*u[i-1,:] + 0.75*u[i+1,:] - 0.15*u[i+2,:] + 0.016666666666667*u[i+3,:]
ux[:,:] = ux[:,:]/dx
def dy(uy, u, ny, dy, order):
# summation-by-parts finite difference operators for first derivatives du/dy
m = ny-1
if order == 2:
uy[:,0]= u[:,1]-u[:,0]
uy[: m] = u[:,m]-u[:,m-1]
for j in range(1, m):
uy[:, j] = 0.5*(u[:, j+1] - u[:, j-1])
uy[:,:] = 1.0/dy*uy[:,:]
if order == 4:
uy[:, 0] = -24.0/17.0*u[:,0] + 59.0/34.0*u[:,1] - 4.0/17.0*u[:, 2] - 3.0/34.0*u[:, 3]
uy[:, 1] = -1.0/2.0*u[:,0] + 1.0/2.0*u[:,2]
uy[:, 2] = 4.0/43.0*u[:,0] - 59.0/86.0*u[:,1] + 59.0/86.0*u[:,3] - 4.0/43.0*u[:,4]
uy[:, 3] = 3.0/98.0*u[:,0] - 59.0/98.0*u[:,2] + 32.0/49.0*u[:,4] - 4.0/49.0*u[:,5]
uy[:, m] = 24.0/17.0*u[:,m] - 59.0/34.0*u[:,m-1] + 4.0/17.0*u[:,m-2] + 3.0/34.0*u[:,m-3]
uy[:, m-1] = 1.0/2.0*u[:,m] - 1.0/2.0*u[:,m-2]
uy[:, m-2] = -4.0/43.0*u[:,m] + 59.0/86.0*u[:,m-1] - 59.0/86.0*u[:,m-3] + 4.0/43.0*u[:,m-4]
uy[:, m-3] = -3.0/98.0*u[:,m] + 59.0/98.0*u[:,m-2] - 32.0/49.0*u[:,m-4] + 4.0/49.0*u[:,m-5]
for j in range(4, m-3):
uy[:, j] = 1.0/12.0*u[:,j-2] - 2.0/3.0*u[:,j-1] + 2.0/3.0*u[:,j+1] - 1.0/12.0*u[:,j+2]
uy[:,:] = 1.0/dy*uy[:,:]
if order == 6:
uy[:,0] = -1.694834962162858*u[:,0] + 2.245634824947698*u[:,1] - 0.055649692295628*u[:,2] - 0.670383570370653*u[:,3]\
- 0.188774952148393*u[:,4] + 0.552135032829910*u[:,5] - 0.188126680800077*u[:,6]
uy[:,1] = -0.434411786832708*u[:,0] + 0.107043134706685*u[:,2] + 0.420172642668695*u[:,3] + 0.119957288069806*u[:,4]\
- 0.328691543801578*u[:,5] + 0.122487487014485*u[:,6] - 0.006557221825386*u[:,7]
uy[:,2] = 0.063307644169533*u[:,0] - 0.629491308812471*u[:,1] + 0.809935419586724*u[:,3] - 0.699016381364484*u[:,4]\
+ 0.850345731199969*u[:,5] - 0.509589652965290*u[:,6] + 0.114508548186019*u[:,7]
uy[:,3] = 0.110198643174386*u[:,0] - 0.357041083340051*u[:,1] - 0.117033418681039*u[:,2] + 0.120870009174558*u[:,4]\
+ 0.349168902725368*u[:,5] - 0.104924741749615*u[:,6] - 0.001238311303608*u[:,7]
uy[:,4] = 0.133544619364965*u[:,0] - 0.438678347579289*u[:,1] + 0.434686341173840*u[:,2] - 0.520172867814934*u[:,3]\
+ 0.049912002176267*u[:,5] + 0.504693510958978*u[:,6] - 0.163985258279827*u[:,7]
uy[:,5] = -0.127754693486067*u[:,0] + 0.393149407857401*u[:,1] - 0.172955234680916*u[:,2] - 0.491489487857764*u[:,3]\
- 0.016325050231672*u[:,4] + 0.428167552785852*u[:,6] - 0.025864364383975*u[:,7] + 0.013071869997141*u[:,8]
uy[:,6] = 0.060008241515128*u[:,0] - 0.201971348965594*u[:,1] + 0.142885356631256*u[:,2] + 0.203603636754774*u[:,3]\
- 0.227565385120003*u[:,4] - 0.590259111130048*u[:,5] + 0.757462553894374*u[:,7] - 0.162184436527372*u[:,8]\
+ 0.018020492947486*u[:,9]
uy[:,7] = 0.009910488565285*u[:,1] - 0.029429452176588*u[:,2] + 0.002202493355677*u[:,3] + 0.067773581604826*u[:,4]\
+ 0.032681945726690*u[:,5] - 0.694285851935105*u[:,6] + 0.743286642396343*u[:,8] - 0.148657328479269*u[:,9]\
+ 0.016517480942141*u[:,10]
uy[:,m-7] =-0.016517480942141*u[:,m-10] + 0.148657328479269*u[:,m-9] - 0.743286642396343*u[:,m-8] + 0.694285851935105*u[:,m-6]\
- 0.032681945726690*u[:,m-5] - 0.067773581604826*u[:,m-4] - 0.002202493355677*u[:,m-3] + 0.029429452176588*u[:,m-2]\
- 0.009910488565285*u[:,m-1]
uy[:,m-6] =-0.018020492947486*u[:,m-9] + 0.162184436527372*u[:,m-8] - 0.757462553894374*u[:,m-7] + 0.590259111130048*u[:,m-5]\
+ 0.227565385120003*u[:,m-4] - 0.203603636754774*u[:,m-3] - 0.142885356631256*u[:,m-2] + 0.201971348965594*u[:,m-1]\
- 0.060008241515128*u[:,m]
uy[:,m-5] =-0.013071869997141*u[:,m-8] + 0.025864364383975*u[:,m-7] - 0.428167552785852*u[:,m-6] + 0.016325050231672*u[:,m-4]\
+ 0.491489487857764*u[:,m-3] + 0.172955234680916*u[:,m-2] - 0.393149407857401*u[:,m-1] + 0.127754693486067*u[:,m]
uy[:,m-4] = 0.163985258279827*u[:,m-7] - 0.504693510958978*u[:,m-6] - 0.049912002176267*u[:,m-5] + 0.520172867814934*u[:,m-3]\
- 0.434686341173840*u[:,m-2] + 0.438678347579289*u[:,m-1] - 0.133544619364965*u[:,m]
uy[:,m-3] = 0.001238311303608*u[:,m-7] + 0.104924741749615*u[:,m-6] - 0.349168902725368*u[:,m-5] - 0.120870009174558*u[:,m-4]\
+ 0.117033418681039*u[:,m-2] + 0.357041083340051*u[:,m-1] - 0.110198643174386*u[:,m]
uy[:,m-2] =-0.114508548186019*u[:,m-7] + 0.509589652965290*u[:,m-6] - 0.850345731199969*u[:,m-5] + 0.699016381364484*u[:,m-4]\
- 0.809935419586724*u[:,m-3] + 0.629491308812471*u[:,m-1] - 0.063307644169533*u[:,m]
uy[:,m-1] = 0.006557221825386*u[:,m-7] - 0.122487487014485*u[:,m-6] + 0.328691543801578*u[:,m-5] - 0.119957288069806*u[:,m-4]\
- 0.420172642668695*u[:,m-3] - 0.107043134706685*u[:,m-2] + 0.434411786832708*u[:,m]
uy[:,m] = 0.188126680800077*u[:,m-6] - 0.552135032829910*u[:,m-5] + 0.188774952148393*u[:,m-4] + 0.670383570370653*u[:,m-3]\
+ 0.055649692295628*u[:,m-2] - 2.245634824947698*u[:,m-1] + 1.694834962162858*u[:,m]
for j in range(8, m-7):
uy[:, j] = -0.016666666666666667*u[:,j-3] + 0.15*u[:,j-2] - 0.75*u[:,j-1] + 0.75*u[:,j+1] - 0.15*u[:,j+2] + 0.01666666666666667*u[:,j+3]
uy[:,:] = (1.0/dy)*uy[:,:];
def dx2d(ux, u, nx, i, j, dx, order):
# summation-by-parts finite difference operators for first derivatives du/dx
m = nx-1
# second order accurate case
if order==2:
# calculate partial derivatives on the boundaries:[0, m] with one-sided difference operators
if i == 0:
ux[:] = (u[1, j, :] - u[0,j, :])/dx
if i == m:
ux[:] = (u[m,j, :] - u[m-1,j, :])/dx
#calculate partial derivatives in the interior:(1:nx-1)
if (i > 0) and (i < m):
ux[:] = (u[i+1, j,:] - u[i-1, j,:])/(2.0*dx)
# fourth order accurate case
if order==4:
#################################################
# calculate partial derivatives on the boundaries:(0,1,2,3, : m-3, m-2, m-1, m)
# with one-sided difference operators
if i == 0:
ux[:] = -24./17*u[0,j,:] + 59./34*u[1, j,:] - 4./17*u[2, j,:] - 3./34*u[3,j,:]
if i == 1:
ux[:] = -1./2*u[0,j,:] + 1./2*u[2, j,:]
if i == 2:
ux[:] = 4./43*u[0,j,:] - 59./86*u[1, j,:] + 59./86*u[3, j,:] - 4./43*u[4,j,:]
if i == 3:
ux[:] = 3./98*u[0,j,:] - 59./98*u[2, j,:] + 32./49*u[4, j,:] - 4./49*u[5,j,:]
if i == m:
ux[:] = 24./17*u[m,j,:] - 59./34*u[m-1, j,:] + 4./17*u[m-2, j,:] + 3./34*u[m-3,j,:]
if i == m-1:
ux[:] = 1./2*u[m,j,:] - 1./2*u[m-2, j,:]
if i == m-2:
ux[:] = -4./43*u[m,j,:] + 59./86*u[m-1, j,:]- 59./86*u[m-3, j,:]+ 4./43*u[m-4,j,:]
if i == m-3:
ux[:] = -3./98*u[m,j,:] + 59./98*u[m-2, j,:]- 32./49*u[m-4, j,:]+ 4./49*u[m-5,j,:]
#------------------------------------------------------------------------------------------------------------------------------
if (i > 3) and (i<m-3):
ux[:] = 0.083333333333333*u[i-2,j,:] - 0.666666666666667*u[i-1,j,:] + 0.666666666666667*u[i+1,j,:] - 0.083333333333333*u[i+2,j,:]
ux[:] = ux[:]/dx
# sixth order accurate case
#################################################
if order==6:
# calculate partial derivatives on the boundaries:(0,1,2,3,4,5,6 : m-6, m-5, m-4, m-3, m-2, m-1, m)
# with one-sided difference operators
if i == 0:
ux[:] = -1.694834962162858*u[0,j,:] + 2.245634824947698*u[1,j,:] - 0.055649692295628*u[2,j,:] - 0.670383570370653*u[3,j,:]\
- 0.188774952148393*u[4,j,:] + 0.552135032829910*u[5,j,:] - 0.188126680800077*u[6,j,:]
if i == 1:
ux[:] = -0.434411786832708*u[0,j,:] + 0.107043134706685*u[2,j,:] + 0.420172642668695*u[3,j,:] + 0.119957288069806*u[4,j,:]\
- 0.328691543801578*u[5,j,:] + 0.122487487014485*u[6,j,:] - 0.006557221825386*u[7,j,:]
if i == 2:
ux[:] = 0.063307644169533*u[0,j,:] - 0.629491308812471*u[1,j,:] + 0.809935419586724*u[3,j,:] - 0.699016381364484*u[4,j,:]\
+ 0.850345731199969*u[5,j,:] - 0.509589652965290*u[6,j,:] + 0.114508548186019*u[7,j,:]
if i == 3:
ux[:] = 0.110198643174386*u[0,j,:] - 0.357041083340051*u[1,j,:] - 0.117033418681039*u[2,j,:] + 0.120870009174558*u[4,j,:]\
+ 0.349168902725368*u[5,j,:] - 0.104924741749615*u[6,j,:] - 0.001238311303608*u[7,j,:]
if i == 4:
ux[:] = 0.133544619364965*u[0,j,:] - 0.438678347579289*u[1,j,:] + 0.434686341173840*u[2,j,:] - 0.520172867814934*u[3,j,:]\
+ 0.049912002176267*u[5,j,:] + 0.504693510958978*u[6,j,:] - 0.163985258279827*u[7,j,:]
if i == 5:
ux[:] = -0.127754693486067*u[0,j,:] + 0.393149407857401*u[1,j,:] - 0.172955234680916*u[2,j,:] - 0.491489487857764*u[3,j,:]\
- 0.016325050231672*u[4,j,:] + 0.428167552785852*u[6,j,:] - 0.025864364383975*u[7,j,:] + 0.013071869997141*u[8,j,:]
if i == 6:
ux[:] = 0.060008241515128*u[0,j,:] - 0.201971348965594*u[1,j,:] + 0.142885356631256*u[2,j,:] + 0.203603636754774*u[3,j,:]\
- 0.227565385120003*u[4,j,:] - 0.590259111130048*u[5,j,:] + 0.757462553894374*u[7,j,:] - 0.162184436527372*u[8,j,:]\
+ 0.018020492947486*u[9,j,:]
if i == 7:
ux[:] = 0.009910488565285*u[1,j,:] - 0.029429452176588*u[2,j,:] + 0.002202493355677*u[3,j,:] + 0.067773581604826*u[4,j,:]\
+ 0.032681945726690*u[5,j,:] - 0.694285851935105*u[6,j,:] + 0.743286642396343*u[8,j,:] - 0.148657328479269*u[9,j,:]\
+ 0.016517480942141*u[10,j,:]
if i == m-7:
ux[:] =-0.016517480942141*u[m-10,j,:] + 0.148657328479269*u[m-9,j,:] - 0.743286642396343*u[m-8,j,:] + 0.694285851935105*u[m-6,j,:]\
- 0.032681945726690*u[m-5,j,:] - 0.067773581604826*u[m-4,j,:] - 0.002202493355677*u[m-3,j,:] + 0.029429452176588*u[m-2,j,:]\
- 0.009910488565285*u[m-1,j,:]
if i == m-6:
ux[:] =-0.018020492947486*u[m-9,j,:] + 0.162184436527372*u[m-8,j,:] - 0.757462553894374*u[m-7,j,:] + 0.590259111130048*u[m-5,j,:]\
+ 0.227565385120003*u[m-4,j,:] - 0.203603636754774*u[m-3,j,:] - 0.142885356631256*u[m-2,j,:] + 0.201971348965594*u[m-1,j,:]\
- 0.060008241515128*u[m,j,:]
if i == m-5:
ux[:] =-0.013071869997141*u[m-8,j,:] + 0.025864364383975*u[m-7,j,:] - 0.428167552785852*u[m-6,j,:] + 0.016325050231672*u[m-4,j,:]\
+ 0.491489487857764*u[m-3,j,:] + 0.172955234680916*u[m-2,j,:] - 0.393149407857401*u[m-1,j,:] + 0.127754693486067*u[m,j,:]
if i == m-4:
ux[:] = 0.163985258279827*u[m-7,j,:] - 0.504693510958978*u[m-6,j,:] - 0.049912002176267*u[m-5,j,:] + 0.520172867814934*u[m-3,j,:]\
- 0.434686341173840*u[m-2,j,:] + 0.438678347579289*u[m-1,j,:] - 0.133544619364965*u[m,j,:]
if i == m-3:
ux[:] = 0.001238311303608*u[m-7,j,:] + 0.104924741749615*u[m-6,j,:] - 0.349168902725368*u[m-5,j,:] - 0.120870009174558*u[m-4,j,:]\
+ 0.117033418681039*u[m-2,j,:] + 0.357041083340051*u[m-1,j,:] - 0.110198643174386*u[m,j,:]
if i == m-2:
ux[:] =-0.114508548186019*u[m-7,j,:] + 0.509589652965290*u[m-6,j,:] - 0.850345731199969*u[m-5,j,:] + 0.699016381364484*u[m-4,j,:]\
- 0.809935419586724*u[m-3,j,:] + 0.629491308812471*u[m-1,j,:] - 0.063307644169533*u[m,j,:]
if i == m-1:
ux[:] = 0.006557221825386*u[m-7,j,:] - 0.122487487014485*u[m-6,j,:] + 0.328691543801578*u[m-5,j,:] - 0.119957288069806*u[m-4,j,:]\
- 0.420172642668695*u[m-3,j,:] - 0.107043134706685*u[m-2,j,:] + 0.434411786832708*u[m,j,:]
if i == m:
ux[:] = 0.188126680800077*u[m-6,j,:] - 0.552135032829910*u[m-5,j,:] + 0.188774952148393*u[m-4,j,:] + 0.670383570370653*u[m-3,j,:]\
+ 0.055649692295628*u[m-2,j,:] - 2.245634824947698*u[m-1,j,:] + 1.694834962162858*u[m,j,:]
if (i > 7) and (i < m-7):
ux[:] = -0.016666666666667*u[i-3,j,:] + 0.15*u[i-2,j,:] - 0.75*u[i-1,j,:] + 0.75*u[i+1,j,:] - 0.15*u[i+2,j,:] + 0.016666666666667*u[i+3,j,:]
ux[:] = ux[:]/dx
def dy2d(uy, u, ny, i, j, dy, order):
# summation-by-parts finite difference operators for first derivatives du/dy
m = ny-1
if order == 2:
if j == 0:
uy[:]= u[i,1,:]-u[i,0,:]
if j == m:
uy[:] = u[i,m,:]-u[i,m-1,:]
if (j > 0) and j < m:
uy[:] = 0.5*(u[i, j+1,:] - u[i, j-1,:])
uy[:] = 1.0/dy*uy[:]
if order == 4:
if j == 0:
uy[:] = -24.0/17.0*u[i,0,:] + 59.0/34.0*u[i,1,:] - 4.0/17.0*u[i, 2,:] - 3.0/34.0*u[i, 3,:]
if j == 1:
uy[:] = -1.0/2.0*u[i,0,:] + 1.0/2.0*u[i,2,:]
if j == 2:
uy[:] = 4.0/43.0*u[i,0,:] - 59.0/86.0*u[i,1,:] + 59.0/86.0*u[i,3,:] - 4.0/43.0*u[i,4,:]
if j == 3:
uy[:] = 3.0/98.0*u[i,0,:] - 59.0/98.0*u[i,2,:] + 32.0/49.0*u[i,4,:] - 4.0/49.0*u[i,5,:]
if j == m:
uy[:] = 24.0/17.0*u[i,m,:] - 59.0/34.0*u[i,m-1,:] + 4.0/17.0*u[i,m-2,:] + 3.0/34.0*u[i,m-3,:]
if j == m-1:
uy[:] = 1.0/2.0*u[i,m,:] - 1.0/2.0*u[i,m-2,:]
if j == m-2:
uy[:] = -4.0/43.0*u[i,m,:] + 59.0/86.0*u[i,m-1,:] - 59.0/86.0*u[i,m-3,:] + 4.0/43.0*u[i,m-4,:]
if j == m-3:
uy[:] = -3.0/98.0*u[i,m,:] + 59.0/98.0*u[i,m-2,:] - 32.0/49.0*u[i,m-4,:] + 4.0/49.0*u[i,m-5,:]
if (j> 3) and (j<m-3):
uy[:] = 1.0/12.0*u[i,j-2,:] - 2.0/3.0*u[i,j-1,:] + 2.0/3.0*u[i,j+1,:] - 1.0/12.0*u[i,j+2,:]
uy[:] = 1.0/dy*uy[:]
if order == 6:
if j == 0:
uy[:] = -1.694834962162858*u[i,0,:] + 2.245634824947698*u[i,1,:] - 0.055649692295628*u[i,2,:] - 0.670383570370653*u[i,3,:]\
- 0.188774952148393*u[i,4,:] + 0.552135032829910*u[i,5,:] - 0.188126680800077*u[i,6,:]
if j == 1:
uy[:] = -0.434411786832708*u[i,0,:] + 0.107043134706685*u[i,2,:] + 0.420172642668695*u[i,3,:] + 0.119957288069806*u[i,4,:]\
- 0.328691543801578*u[i,5,:] + 0.122487487014485*u[i,6,:] - 0.006557221825386*u[i,7,:]
if j == 2:
uy[:] = 0.063307644169533*u[i,0,:] - 0.629491308812471*u[i,1,:] + 0.809935419586724*u[i,3,:] - 0.699016381364484*u[i,4,:]\
+ 0.850345731199969*u[i,5,:] - 0.509589652965290*u[i,6,:] + 0.114508548186019*u[i,7,:]
if j == 3:
uy[:] = 0.110198643174386*u[i,0,:] - 0.357041083340051*u[i,1,:] - 0.117033418681039*u[i,2,:] + 0.120870009174558*u[i,4,:]\
+ 0.349168902725368*u[i,5,:] - 0.104924741749615*u[i,6,:] - 0.001238311303608*u[i,7,:]
if j == 4:
uy[:] = 0.133544619364965*u[i,0,:] - 0.438678347579289*u[i,1,:] + 0.434686341173840*u[i,2,:] - 0.520172867814934*u[i,3,:]\
+ 0.049912002176267*u[i,5,:] + 0.504693510958978*u[i,6,:] - 0.163985258279827*u[i,7,:]
if j == 5:
uy[:] = -0.127754693486067*u[i,0,:] + 0.393149407857401*u[i,1,:] - 0.172955234680916*u[i,2,:] - 0.491489487857764*u[i,3,:]\
- 0.016325050231672*u[i,4,:] + 0.428167552785852*u[i,6,:] - 0.025864364383975*u[i,7,:] + 0.013071869997141*u[i,8,:]
if j == 6:
uy[:] = 0.060008241515128*u[i,0,:] - 0.201971348965594*u[i,1,:] + 0.142885356631256*u[i,2,:] + 0.203603636754774*u[i,3,:]\
- 0.227565385120003*u[i,4,:] - 0.590259111130048*u[i,5,:] + 0.757462553894374*u[i,7,:] - 0.162184436527372*u[i,8,:]\
+ 0.018020492947486*u[i,9,:]
if j == 7:
uy[:] = 0.009910488565285*u[i,1,:] - 0.029429452176588*u[i,2,:] + 0.002202493355677*u[i,3,:] + 0.067773581604826*u[i,4,:]\
+ 0.032681945726690*u[i,5,:] - 0.694285851935105*u[i,6,:] + 0.743286642396343*u[i,8,:] - 0.148657328479269*u[i,9,:]\
+ 0.016517480942141*u[i,10,:]
if j == m-7:
uy[:] =-0.016517480942141*u[i,m-10,:] + 0.148657328479269*u[i,m-9,:] - 0.743286642396343*u[i,m-8,:] + 0.694285851935105*u[i,m-6,:]\
- 0.032681945726690*u[i,m-5,:] - 0.067773581604826*u[i,m-4,:] - 0.002202493355677*u[i,m-3,:] + 0.029429452176588*u[i,m-2,:]\
- 0.009910488565285*u[i,m-1,:]
if j == m-6:
uy[:] =-0.018020492947486*u[i,m-9,:] + 0.162184436527372*u[i,m-8,:] - 0.757462553894374*u[i,m-7,:] + 0.590259111130048*u[i,m-5,:]\
+ 0.227565385120003*u[i,m-4,:] - 0.203603636754774*u[i,m-3,:] - 0.142885356631256*u[i,m-2,:] + 0.201971348965594*u[i,m-1,:]\
- 0.060008241515128*u[i,m,:]
if j == m-5:
uy[:] =-0.013071869997141*u[i,m-8,:] + 0.025864364383975*u[i,m-7,:] - 0.428167552785852*u[i,m-6,:] + 0.016325050231672*u[i,m-4,:]\
+ 0.491489487857764*u[i,m-3,:] + 0.172955234680916*u[i,m-2,:] - 0.393149407857401*u[i,m-1,:] + 0.127754693486067*u[i,m,:]
if j == m-4:
uy[:] = 0.163985258279827*u[i,m-7,:] - 0.504693510958978*u[i,m-6,:] - 0.049912002176267*u[i,m-5,:] + 0.520172867814934*u[i,m-3,:]\
- 0.434686341173840*u[i,m-2,:] + 0.438678347579289*u[i,m-1,:] - 0.133544619364965*u[i,m,:]
if j == m-3:
uy[:] = 0.001238311303608*u[i,m-7,:] + 0.104924741749615*u[i,m-6,:] - 0.349168902725368*u[i,m-5,:] - 0.120870009174558*u[i,m-4,:]\
+ 0.117033418681039*u[i,m-2,:] + 0.357041083340051*u[i,m-1,:] - 0.110198643174386*u[i,m,:]
if j == m-2:
uy[:] =-0.114508548186019*u[i,m-7,:] + 0.509589652965290*u[i,m-6,:] - 0.850345731199969*u[i,m-5,:] + 0.699016381364484*u[i,m-4,:]\
- 0.809935419586724*u[i,m-3,:] + 0.629491308812471*u[i,m-1,:] - 0.063307644169533*u[i,m,:]
if j == m-1:
uy[:] = 0.006557221825386*u[i,m-7,:] - 0.122487487014485*u[i,m-6,:] + 0.328691543801578*u[i,m-5,:] - 0.119957288069806*u[i,m-4,:]\
- 0.420172642668695*u[i,m-3,:] - 0.107043134706685*u[i,m-2,:] + 0.434411786832708*u[i,m,:]
if j == m:
uy[:] = 0.188126680800077*u[i,m-6,:] - 0.552135032829910*u[i,m-5,:] + 0.188774952148393*u[i,m-4,:] + 0.670383570370653*u[i,m-3,:]\
+ 0.055649692295628*u[i,m-2,:] - 2.245634824947698*u[i,m-1,:] + 1.694834962162858*u[i,m,:]
if (j> 7) and (j<m-7):
uy[:] = -0.016666666666666667*u[i,j-3,:] + 0.15*u[i,j-2,:] - 0.75*u[i,j-1,:] + 0.75*u[i,j+1,:] - 0.15*u[i,j+2,:] + 0.01666666666666667*u[i,j+3,:]
uy[:] = (1.0/dy)*uy[:]