RIVER: DG-Fourier Multi-step SSP method for the 2-D advection equation

Purpose

• The package "River" solves the 2-D advection equation in a rectangular domain. A nodal discontinuous Galerkin method was used for the spatial discretization in $x$, and a pseudo-Fourier method was used in $y$. The temporal discretization are multi-step strong stability-preserving (SSP) schemes. It is partially based on packages "Sea" and "Ion".

Specifications

• Name: River.
• Author: Feng Chen.
• Finishing date: 11/26/2012.
• Languages: Fortran 90.
• Required libraries: BLAS, LAPACK.

Simple Example

• Equation: $$\begin{equation} \begin{aligned} & \Psi_t + \boldsymbol{u} \cdot \nabla \Psi = f, \\ & \Psi(x,y,0) = \Psi_0(x,y), \\ & \Psi \text{ periodic in } x, \\ & \Psi \text{ no boundary condition in } y, \end{aligned} \end{equation}$$ where $\boldsymbol{u}$ satisfies $$\begin{equation} \nabla \cdot \boldsymbol{u} = 0, \quad \boldsymbol{u} \cdot \boldsymbol{n} |_{y=\pm 1}= 0. \end{equation}$$
• Parameters: $$\begin{equation} \Omega =(-\pi,\pi)\times (-1,1), \, T=1, \, \delta t = 0.0001, \, N_x = 64, \, N_y = 10, \, K_y = 10. \end{equation}$$
• Exact solution and input functions: $$\begin{equation} \begin{aligned} & \Psi(x,y,t) = \cos(x+t) \sin(y+t),\\ & u_1(x,y,t) = \pi \cos(x+t) \cos(\pi y), \\ &u_2(x,y,t) = \sin(x+t) \sin(\pi y), \end{aligned} \end{equation}$$ then $f(x,y)$ is calculated accordingly.

Quick Start

• Compiling and running:

  cd ./River
  make library
  gfortran River_Main.cu -llibrary -llapack -lblas
  ./a.out
  

• Output:

  Order 2 step 2 scheme was used.
   Max error of f at step           0 :   0.0000000000000000     
   Max error of f at step        1000 :  9.98779896381751797E-005
   Max error of f at step        2000 :  1.94693442869153799E-004
   Max error of f at step        3000 :  2.73449928127500463E-004
   Max error of f at step        4000 :  3.29979540826086382E-004
   Max error of f at step        5000 :  3.62972114978599159E-004
   Max error of f at step        6000 :  3.74585022890339125E-004
   Max error of f at step        7000 :  3.72236244570209318E-004
   Max error of f at step        8000 :  3.67233888585664914E-004
   Max error of f at step        9000 :  3.52135748568682683E-004
   Max error of f at step       10000 :  3.52689697006920710E-004
  ...
  

• CPU: Intel(R) Xeon(R) CPU X5550 @2.67GHz.
• OS: CentOS release 6.4 (Final).
• Compiler: gfortran 4.5.1.

References

• Gottlieb, S. and Shu, C. W. and Tadmor, E. Strong stability-preserving high-order time discretization methods, SIAM Review, No. 43 (2001).

Code Highlights

!<
!! RiverTime_Solo propagates one step forward
!! using the multi-step SSP scheme: first-order 
!! and second-order scheme are illustrated.
!<
subroutine RiverTime_Solo(co, ct, T, SI, SF, LI)
 !! order of the scheme
 integer, intent(in) :: co
 !! local time
  real(kind=8), intent(in) :: ct
  !! temporal discretization module
 type(AtomTime) :: T
 !! dynamics variables storing the solution
 type(RiverIce) :: LI
 !! spatial discretization module (y)
 type(domain), intent(in) :: SF
 !! spatial discretization module (x) 
 type(IonSpace1D), intent(in) :: SI
 integer :: io

 if (co .eq. 0) then        !! initialization
  LI%f(:,:,:,0) = LI%fi 
  LI%f(:,:,:,1) = LI%fi 
 elseif (co .eq. 1) then    !! first-order
  call RiverTime_Solo_Flux(co, ct, T, SI, SF, LI)
  LI%f(:,:,:,1) = LI%f(:,:,:,0) + T%dt*LI%fb(:,:,:,0) 
 elseif (co .eq. 2) then    !! second-order
   call RiverTime_Solo_Flux(co, ct, T, SI, SF, LI)
   LI%f(:,:,:,2) = (4d0/5d0)*LI%f(:,:,:,1) + &
    (1d0/5d0)*LI%f(:,:,:,0) + &
    T%dt*(8d0/5d0)*LI%fb(:,:,:,1) + &
    T%dt*(-2d0/5d0)*LI%fb(:,:,:,0)
 endif
end subroutine RiverTime_Solo