!******************************************************************** ! PMGsolver ! A parallel multigrid solver for the 2D linear problems ! arising in the plasma fluid model GDB (see B. Zhu, et. al., ! GDB: a global 3D two-fluid code for plasma turbulence and ! transport in the tokamak edge region, J. of Comp. Phys 2017). ! ! These problems are: ! 1. A Poisson solver to obtain the electrostatic potential phi ! from the vorticity w ! div(lambda*grad(u))=rho ! where lambda=plasma density, u=phi and rho=w. ! 2. An inhomogeneous modified Helmholtz equation to obtain the ! magnetic flux function psi from the variable rpsi which ! contains an electron inertia contribution ! u-ksq*L(u)=rho ! where u=psi, rho=rpsi and ksq=(electron skin depth)^2 ! In the routines below we use the suffix 'ps' to indicate ! routines to obtain psi ! 3. Higher order variants of the (modified) Helmholtz equation ! arising from implied hyperdiffusion of order 2m ! u+((-1)^(m+1))*Diff*(L^2m)(u)=rho ! where Diff is the diffusion coefficient and L is the 2D ! Laplacian operator, although a non-Laplacian-based diffusion ! operator is also available respectively. For more details ! see M. Francisquez, et. al., Multigrid treatment of implicit ! continuum diffusion, J. of Comp. Phys 2017) ! 4. A simple Poisson solver ! div(grad(u))=rho ! ! In the routines below we use the suffix 'ph' to indicate ! routines to obtain phi, 'ps' for psi and 'hd' for hyperdiffusion. ! ! Though these are 2D solvers, we allow for a 3rd dimension so that ! PMGsolver effetively supports solving many xy planes of ! independent 2D problems. ! ! A program, pmg4gdb.F90, has been included to show the use of some ! of these solvers with a test function. ! ! Manaure Francisquez ! mana@dartmouth.edu ! 2017 ! !******************************************************************** ! IMPORTANT NOTES ! i) Not all solvers support the same boundary conditions ! ii) Modified Helmholtz relaxation has damped relaxation commented out ! iii) Some restriction and prolongation operators have been removed ! for simplicity. I believe IR=0, ISR=1, IPRO=2 are the only options ! available in this file. ! ASSUMPTIONS ! a) Periodic along y and not periodic along x ! b) The length of the simulation domain is defined as 2*pi times ! a factor specified in the input file (fLx along x) ! c) The problem, the grids and the MPI decomposition are such that ! only two subdomains along x and/or y can be unified in a single ! coarsening. ! ! Poisson ! 1. Each process has at least nxmin and nymin points ! along x and y, respectively. ! 2. nxmin,nymin >= 4 ! 3. smallest grid has >=4 points along x ! 4. Boundary condition on the right (along x) is Dirichlet(? check) ! 5. The boundary conditions of lambda are assumed even ! 6. Use boundary linear extrapolation when restricting lambda ! ! Hyperdiffusion ! 1. Hyperdiffusion may be applied to more than one quantity ! (e.g. density, temperature, etc.), and a different multigrid ! parameter set is given for each. nqhd = number of quantities ! hyperdiffusion is applied to. This variable is inferred from ! the size of iDiff, and the code looks for nqhd parameter sets ! in the input file. ! 2. HDop: type of diffusion, some options are (L=Laplacian) ! =2 u + Diff*(L^2)u=rho ! =3 u - Diff*(L^3)u=rho ! =4 u + Diff*(L^2)u=rho split into 2 eqns ! =6 u - Diff*(L^3)u=rho split into 3 eqns ! =8 u + Diffx*(d^4 u/dx^4) + Diffy*(d^4 u/dy^4)=rho ! =12 u - Diffx*(d^6 u/dx^6) - Diffy*(d^6 u/dy^6)=rho !********************************************************************