LAKE: DG-Fourier Multi-stage SSP method for the 2-D advection equation

Purpose

• The package "Lake" solves the 2-D advection equation in a rectangular domain. A Fourier psuedospectral method was used for the spatial discretization in $x$, and a nodal discontinuous Galerkin method was used for $y$. The temporal discretization are one-step multi-stage explicit schemes. It is partially based on packages "Sea" and "Ion".

Specifications

• Name: Lake.
• Author: Feng Chen.
• Finishing date: 06/26/2012.
• 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: $$\begin{equation} \Omega =(-\pi,\pi)\times (-1,1), \, T=10, \, \delta t = 0.001, \, N_x = 64, \, N_y = 10, \, K_y = 10. \end{equation}$$
• 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 $f(x,y,t)$ is calculated accordingly.

Quick Start

• Compiling and running:

  cd ./Lake
  make library
  gfortran Lake_Main.cu -llibrary -llapack -lblas
  ./a.out
  

• Output:

  Order 3 Runge Kutta was used.
   at step           0 max error=   0.0000000000000000     
   at step        2000 max error=  2.32462854521386930E-009
   at step        4000 max error=  2.56192570491364791E-008
   at step        6000 max error=  6.41112042915753522E-008
   at step        8000 max error=  1.24243067334273150E-007
   at step       10000 max error=  1.15325957505962862E-007
  ...
  

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

References

• Jan S. Hesthaven and Tim Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008.
• The third-order Runge Kutta Method. Link.

Code Highlights

!<
!! calculate the rhs of the linear system
!<
subroutine LakeSpaceTime_RHS(SI, SF, rhs, flux, Psi, fsrc)  
 type(domain), intent(in) ::  SF   
 type(IonSpace1D), intent(in) :: SI 
 real(kind=8), dimension(SI%n,SF%ndof), intent(in) :: Psi, fsrc
 real(kind=8), dimension(SI%n,SF%ndof) :: rhs
 real(kind=8), dimension(SI%n,SF%ndof,2) :: flux 
 integer :: j, k  

 ! differentiation in y
 do k = 1, SI%n
  call NDG_RHS(SF, rhs(k,:), flux(k,:,2), Psi(k,:))
 end do
 flux(:,:,2) = rhs

 ! differentiation in x
 do j = 1, SF%ndof
  ! transform to k-space
  call rfft1d(SI%n, flux(:,j,1), SI%wk, 0)
  ! differentiation
  call fodxcf(SI%n, flux(:,j,1), SI%sc)  
  ! transform to real space
  call rfft1d(SI%n, flux(:,j,1), SI%wk, 1)
 end do 
 ! pseudo-Fourier, no integration by parts.
 flux(:,:,1) = -flux(:,:,1)  


 ! add up all the rhs
   rhs = flux(:,:,1) + flux(:,:,2) + fsrc 
end subroutine LakeSpaceTime_RHS