| 1 | C Last change: C 1 Mar 2001 12:27 pm |
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| 2 | program Landslide_Tadmor |
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| 3 | c |
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| 4 | c Stiff source term solution using the second order |
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| 5 | c central scheme of Landslide and the modifications |
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| 6 | c suggested by Liotta, Romano and Russo, SIAM |
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| 7 | c J. Numer. Anal. 38(4), 1337-1356, 2000. |
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| 8 | c This is the Randau scheme. |
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| 9 | c |
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| 10 | c============================================================ |
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| 11 | implicit none |
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| 12 | integer md,nxmax,nxd, mn |
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| 13 | parameter (md=3, nxmax=1601, nxd=nxmax+2*md, mn=2) |
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| 14 | real*8 u(nxd,mn),x(nxd,2),z(nxd,2),dz(nxd,2) |
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| 15 | |
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| 16 | integer nx, nxx, io, i, isteps |
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| 17 | real*8 toll,cfl,theta,g,dx,dtmax,Tout,Tfinal,tc,a,tau,B,t0, |
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| 18 | . p,s,zeta_0,zeta_1,zeta,u0,version,hl,hr |
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| 19 | |
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| 20 | open(11,file='data.w') |
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| 21 | open(12,file='data.t') |
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| 22 | open(13,file='data.h') |
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| 23 | open(14,file='data.uh') |
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| 24 | open(15,file='data.n') |
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| 25 | open(16,file='data.x') |
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| 26 | |
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| 27 | open(1,file='MacDonald.bed') |
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| 28 | c |
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| 29 | c Parameters |
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| 30 | c |
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| 31 | version = 1.0 |
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| 32 | nx = 101 |
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| 33 | nxx = nx+2*md |
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| 34 | if (nx.gt.nxmax) write(*,*)'nx too large' |
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| 35 | toll = 1.d-6 |
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| 36 | cfl = 0.2d0 |
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| 37 | theta = 1.0d0 |
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| 38 | g = 9.81d0 |
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| 39 | hl=10.0d0 |
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| 40 | hr = 0.d0 |
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| 41 | dx = 10000.d0/dble(nx-1) |
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| 42 | dtmax = 0.5d0 |
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| 43 | Tout = 1.0d0 |
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| 44 | Tfinal = 1500.0d0 |
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| 45 | |
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| 46 | t0 = 0.d0 |
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| 47 | |
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| 48 | c |
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| 49 | c Setup Terrain |
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| 50 | c |
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| 51 | c ---- Non-uniform terrain from MacDonald |
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| 52 | io = 1 |
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| 53 | do i = 1,nxx |
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| 54 | read(1,*)x(i,io),z(i,io) |
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| 55 | z(i,io) = z(i,io)*5.d0 |
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| 56 | end do |
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| 57 | do i = 2,nxx-1 |
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| 58 | dz(i,io) = (z(i+1,io) - z(i-1,io))/2.d0/dx |
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| 59 | end do |
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| 60 | dz(1,io) = dz(2,io) |
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| 61 | dz(nxx,io) = dz(nxx-1,io) |
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| 62 | |
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| 63 | io = 2 |
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| 64 | do i = 1,nxx |
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| 65 | x(i,io) = x(i,io-1) |
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| 66 | z(i,io) = z(i,io-1) |
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| 67 | dz(i,io) = dz(i,io-1) |
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| 68 | end do |
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| 69 | |
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| 70 | c |
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| 71 | c Initial Conditions |
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| 72 | c |
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| 73 | c ---- Rectangular pulse initial condition |
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| 74 | do i = md+1, nx+md |
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| 75 | if (x(i,io).lt.100.d0.or.x(i,io).gt.200.d0) then |
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| 76 | u(i,1) = hr |
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| 77 | else |
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| 78 | u(i,1) = hl |
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| 79 | endif |
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| 80 | u(i,2) = 0.0d0 |
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| 81 | end do |
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| 82 | |
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| 83 | c |
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| 84 | c Setup for Time looping |
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| 85 | c |
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| 86 | |
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| 87 | tc = 0.0d0 |
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| 88 | call dump_solution(u,z,nx,tc,isteps) |
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| 89 | c |
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| 90 | c Evolve |
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| 91 | c |
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| 92 | call momentum_mass(nx,dx,cfl,g,theta,Tfinal, |
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| 93 | . u,dtmax,toll,z,dz,Tout,isteps) |
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| 94 | |
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| 95 | write(15,*)isteps,nxx,version |
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| 96 | do i=1,nxx |
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| 97 | write(16,*) x(i,1) |
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| 98 | enddo |
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| 99 | |
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| 100 | |
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| 101 | c |
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| 102 | c Close down |
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| 103 | c |
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| 104 | close(1) |
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| 105 | close(2) |
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| 106 | close(3) |
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| 107 | close(4) |
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| 108 | close(5) |
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| 109 | close(8) |
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| 110 | |
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| 111 | stop |
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| 112 | end |
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| 113 | |
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| 114 | c=============================================================== |
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| 115 | c The evolver |
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| 116 | c=============================================================== |
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| 117 | subroutine momentum_mass(nx,dx,cfl,g,theta,tf,u,dtmax,toll, |
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| 118 | . z,dz,Tout,isteps) |
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| 119 | c |
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| 120 | c INPUT nx # of cells in x-direction |
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| 121 | c dx: step sizes in x-direction |
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| 122 | c cfl: CFL # g: Acceleration due to gravity |
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| 123 | c tf: final time |
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| 124 | c theta=1: MM1 limiter; =2: MM2 limiter; >2: UNO limiter. |
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| 125 | c u: initial cell averages of conservative variables. |
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| 126 | c Supply entries of u((md+1):(nx+md),mn) |
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| 127 | c OUTPUT u: cell averages at final time 'tf' |
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| 128 | c REMARK 1. Reset 'ndx' to adjust array dimensions. |
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| 129 | c 2. Padded to each side of the computational domain are |
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| 130 | c 'md' ghost cells, average values on which are |
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| 131 | c assigned by boundary conditions. |
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| 132 | c |
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| 133 | implicit none |
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| 134 | integer md,nxmax,nxd, mn |
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| 135 | parameter (md=3, nxmax=1601, nxd=nxmax+2*md, mn=2) |
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| 136 | real*8 u(nxd,mn), ux(nxd,mn) |
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| 137 | real*8 f(nxd,mn), fx(nxd,mn) |
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| 138 | real*8 v(nxd), du(nxd,2), df(nxd,2), dz(nxd,2), u2(nxd,2), |
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| 139 | . u3(nxd,2), h(nxd), z(nxd,2), hz(nxd) |
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| 140 | |
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| 141 | integer nx, nxx, io, i, isteps |
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| 142 | real*8 toll,cfl,theta,g,dx,dt,Tout,Tfinal,tc |
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| 143 | |
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| 144 | integer istop,iout,nt,ii,oi,i2,m |
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| 145 | real*8 dtmax, tf,a,b,xmin,t,xmic,Tleft,em_x,dtcdx2,dtcdx3 |
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| 146 | real*8 den,xmt,pre, Tnext, em_old |
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| 147 | real*8 aterm1 ,term1, aterm2 ,term2, aterm3 ,term3 |
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| 148 | c |
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| 149 | xmin(a,b) = 0.5d0*(dsign(1.d0,a) |
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| 150 | . + dsign(1.d0,b))*dmin1(dabs(a),dabs(b)) |
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| 151 | xmic(t,a,b) = xmin(t*xmin(a,b), 0.5d0*(a+b) ) |
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| 152 | c |
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| 153 | c---------------------------------------------------------- |
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| 154 | c |
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| 155 | em_old = 1.0d0 |
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| 156 | tc = 0.d0 |
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| 157 | istop = 0 |
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| 158 | c iplot = 100 |
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| 159 | isteps = 0 |
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| 160 | iout = 0 |
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| 161 | nt =0 |
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| 162 | Tleft = dmin1(Tout,tf) |
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| 163 | Tnext = dmin1(Tout,tf) |
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| 164 | dt = dmin1(dtmax,Tleft) |
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| 165 | nxx = nx + 2*md |
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| 166 | c |
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| 167 | do while (.true.) |
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| 168 | nt = nt+1 |
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| 169 | do ii = 1, 2 |
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| 170 | io=ii-1 |
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| 171 | oi = 1-io |
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| 172 | i2 = 3-ii |
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| 173 | c |
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| 174 | c Naive Boundary Conditions |
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| 175 | c |
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| 176 | do i = 1, md |
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| 177 | u(i,1) = 0.d0 |
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| 178 | u(nx+i,1) = 0.d0 |
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| 179 | u(i,2) = 0.d0 |
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| 180 | u(nx+i,2) = 0.d0 |
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| 181 | end do |
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| 182 | c |
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| 183 | c Compute f and maximum wave speeds 'em_x' |
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| 184 | c see (2.1) and (4.3) |
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| 185 | c |
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| 186 | call swflux(f,u,1,nxx,em_x,g,toll) |
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| 187 | c |
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| 188 | c Compute numerical derivatives 'ux', 'fx'. |
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| 189 | c see (3.1) and (4.1) |
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| 190 | c |
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| 191 | call derivative(u(1,1),ux(1,1),nxx,theta) |
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| 192 | call derivative(u(1,2),ux(1,2),nxx,theta) |
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| 193 | |
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| 194 | call derivative(f(1,1),fx(1,1),nxx,theta) |
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| 195 | call derivative(f(1,2),fx(1,2),nxx,theta) |
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| 196 | |
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| 197 | c |
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| 198 | c Compute time step size according to the input CFL # |
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| 199 | c |
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| 200 | if(ii.eq.1) then |
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| 201 | dt = dmin1(cfl*dx/em_x,dtmax,Tout) |
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| 202 | if( ( tc + 2.d0*dt ) .ge. Tnext ) then |
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| 203 | dt = 0.5d0*(Tnext - tc ) |
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| 204 | iout = 1 |
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| 205 | Tnext = Tnext+Tout |
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| 206 | end if |
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| 207 | if (Tnext.gt.tf ) then |
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| 208 | istop = 1 |
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| 209 | endif |
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| 210 | end if |
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| 211 | |
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| 212 | dtcdx2 = dt/dx/2.d0 |
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| 213 | dtcdx3 = dt/dx/3.d0 |
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| 214 | c |
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| 215 | c Compute the flux values of f at half time step, |
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| 216 | c the conservative values at the half and third time steps. |
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| 217 | c see (2.15) and (2.16). |
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| 218 | do i = 3, nxx-2 |
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| 219 | c Half time step |
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| 220 | u2(i,1) = u(i,1) - dtcdx2*fx(i,1) |
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| 221 | u2(i,2) = u(i,2) - dtcdx2*fx(i,2) |
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| 222 | c Third Step |
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| 223 | u3(i,1) = u(i,1) - dtcdx3*fx(i,1) |
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| 224 | u3(i,2) = u(i,2) - dtcdx3*fx(i,2) |
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| 225 | if(u3(i,1).lt.toll)then |
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| 226 | u3(i,1) = 0.d0 |
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| 227 | end if |
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| 228 | enddo |
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| 229 | |
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| 230 | call swflux(f,u2,3,nxx-2,em_x,g,toll) |
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| 231 | |
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| 232 | c |
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| 233 | c Compute the values of 'u' at the next time level. see (2.16). |
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| 234 | c Continuity equation |
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| 235 | m = 1 |
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| 236 | aterm1 = 0.0d0 |
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| 237 | aterm2 = 0.0d0 |
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| 238 | aterm3 = 0.0d0 |
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| 239 | |
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| 240 | do i = md + 1 - io, nx + md - io |
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| 241 | term1 = (0.250d0 * ( u(i,m) + u(i+1,m) ) |
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| 242 | . + 0.0625d0 * ( ux(i,m) - ux(i+1,m) ) |
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| 243 | . + dtcdx2 * ( f(i,m) - f(i+1,m) ) )*2.d0 |
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| 244 | v(i) = term1 |
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| 245 | aterm1 = dmax1(aterm1,abs(term1)) |
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| 246 | end do |
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| 247 | do i = md + 1, nx + md |
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| 248 | u(i,m) = v(i-io) |
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| 249 | end do |
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| 250 | |
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| 251 | c |
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| 252 | c Momentum equation |
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| 253 | m = mn |
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| 254 | |
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| 255 | do i = md + 1 - io, nx + md - io |
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| 256 | term2 = (0.250d0 * ( u(i,m) + u(i+1,m) ) |
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| 257 | . + 0.0625d0 * ( ux(i,m) - ux(i+1,m) ) |
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| 258 | . + dtcdx2 * ( f(i,m) - f(i+1,m) ) )*2.d0 |
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| 259 | v(i) = term2 |
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| 260 | aterm2 = dmax1(aterm2,abs(term2)) |
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| 261 | term3 = 0.5d0*dt*g*( u2(i,1)*dz(i,ii) + |
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| 262 | . u2(i+1,1)*dz(i+1,ii)) |
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| 263 | aterm3 = dmax1(aterm3,abs(term3)) |
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| 264 | v(i) = v(i) - term3 |
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| 265 | end do |
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| 266 | |
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| 267 | do i = md + 1, nx + md |
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| 268 | u(i,m) = v(i-io) |
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| 269 | c |
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| 270 | c Friction term |
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| 271 | c u(i,m) = u(i,m)*exp(-0.001d0*dt) |
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| 272 | end do |
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| 273 | c |
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| 274 | tc = tc + dt |
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| 275 | end do |
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| 276 | |
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| 277 | if(iout.eq.1) then |
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| 278 | call dump_solution(u,z,nx,tc,isteps) |
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| 279 | iout = 0 |
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| 280 | endif |
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| 281 | if(istop.eq.1) go to 1001 |
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| 282 | end do |
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| 283 | c |
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| 284 | 1001 write(*,*)isteps |
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| 285 | |
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| 286 | return |
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| 287 | |
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| 288 | end |
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| 289 | |
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| 290 | c=========================================================== |
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| 291 | c Shallow Water Flux |
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| 292 | c=========================================================== |
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| 293 | subroutine swflux(f,q,n0,n1,em_x,g,toll) |
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| 294 | implicit real*8 (a-h) |
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| 295 | implicit real*8 (o-z) |
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| 296 | parameter (md=3, nxmax=1601, nxd=nxmax+2*md, mn=2) |
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| 297 | real*8 f(nxd,mn),q(nxd,mn) |
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| 298 | integer n0,n1 |
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| 299 | |
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| 300 | |
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| 301 | em_x = 1.d-15 |
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| 302 | do i = n0,n1 |
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| 303 | h = q(i,1) |
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| 304 | uh = q(i,2) |
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| 305 | if(h.le.toll)then |
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| 306 | u = 0.d0 |
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| 307 | h = 0.0d0 |
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| 308 | uh = 0.0d0 |
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| 309 | else |
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| 310 | u = uh / h |
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| 311 | endif |
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| 312 | if (abs(u).gt.1.0d3) then |
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| 313 | u = dsign(1.0d0,u)*1.0d3 |
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| 314 | uh = u*h |
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| 315 | end if |
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| 316 | cvel = dsqrt( g*h ) |
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| 317 | em_x = dmax1( em_x, dabs(u) + cvel ) |
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| 318 | f(i,1) = uh |
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| 319 | f(i,2) = uh*u + 0.5d0*g*h**2 |
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| 320 | c q(i,1) = h |
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| 321 | c q(i,2) = uh |
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| 322 | enddo |
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| 323 | |
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| 324 | return |
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| 325 | end |
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| 326 | c=========================================================== |
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| 327 | c Calculate limited derivatives |
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| 328 | c=========================================================== |
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| 329 | subroutine derivative(q,qx,n,theta) |
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| 330 | implicit real*8 (a-h) |
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| 331 | implicit real*8 (o-z) |
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| 332 | |
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| 333 | parameter (md=3, nxmax=1601, nxd=nxmax+2*md, mn=2) |
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| 334 | real*8 dq(nxd,2) |
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| 335 | real*8 q(n),qx(n) |
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| 336 | |
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| 337 | xmin(a,b) = 0.5d0*(dsign(1.d0,a) |
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| 338 | . + dsign(1.d0,b))*dmin1(dabs(a),dabs(b)) |
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| 339 | xmic(z,a,b) = xmin(z*xmin(a,b), 0.5d0*(a+b) ) |
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| 340 | |
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| 341 | |
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| 342 | do i = 1,n-1 |
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| 343 | dq(i,1) = q(i+1) - q(i) |
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| 344 | end do |
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| 345 | do i = 1,n-2 |
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| 346 | dq(i,2) = dq(i+1,1) - dq(i,1) |
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| 347 | end do |
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| 348 | if( theta .lt. 2.5d0 ) then |
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| 349 | do i = 3,n-2 |
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| 350 | qx(i) = xmic( theta, dq(i-1,1), dq(i,1) ) |
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| 351 | end do |
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| 352 | else |
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| 353 | do i = 3,n-2 |
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| 354 | qx(i) = xmin(dq(i-1,1) |
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| 355 | . + 0.5d0*xmin(dq(i-2,2),dq(i-1,2)), |
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| 356 | . dq(i,1) - 0.5d0*xmin(dq(i-1,2),dq(i,2))) |
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| 357 | end do |
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| 358 | end if |
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| 359 | return |
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| 360 | end |
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| 361 | |
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| 362 | |
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| 363 | |
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| 364 | c=========================================================== |
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| 365 | c Output Routine |
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| 366 | c=========================================================== |
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| 367 | subroutine dump_solution(u,z,nx,tc,isteps) |
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| 368 | implicit real*8 (a-h) |
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| 369 | implicit real*8 (o-z) |
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| 370 | parameter (md=3, nxmax=1601, nxd=nxmax+2*md, mn=2) |
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| 371 | real*8 u(nxd,mn),z(nxd,2) |
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| 372 | |
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| 373 | write(12,*)tc |
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| 374 | io = 1 |
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| 375 | i = 2 |
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| 376 | isteps = isteps + 1 |
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| 377 | write(*,*)isteps, tc |
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| 378 | |
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| 379 | do i = 1, nx + 2*md |
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| 380 | write(13,200)u(i,1) |
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| 381 | write(14,200)u(i,2) |
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| 382 | write(11,200)z(i,io) + u(i,1) |
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| 383 | 200 format(e20.8) |
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| 384 | end do |
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| 385 | |
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| 386 | return |
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| 387 | end |
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