C++ Example Problems

Parallel Matrix-Free Example: kin_heat2D_nonlin_p

As an illustration using KINSOL for the solution of nonlinear systems in parallel, we give a sample program called kin_heat2D_nonlin_p.cpp. It uses the KINSOL fixed-point (KIN_FP) iteration with Anderson Acceleration and the MPI parallel vector for the solution of a steady-state 2D heat equation with an additional nonlinear term defined by \(c(u)\):

\[b = \nabla \cdot (D \nabla u) + c(u) \quad \text{in} \quad \mathcal{D} = [0,1] \times [0,1]\]

where \(D\) is a diagonal matrix with entries \(k_x\) and \(k_y\) for the diffusivity in the \(x\) and \(y\) directions, respectively. The boundary conditions are

\[u(0,y) = u(1,y) = u(x,0) = u(x,1) = 0.\]

We chose the analytical solution to be

\[u_{\text{exact}} = u(x,y) = \sin^2(\pi x) \sin^2(\pi y)\]

Hence, we define the static term \(b\) as follows

\[b = k_x 2 \pi^2 (\cos^2(\pi x) - \sin^2(\pi x)) \sin^2(\pi y) + k_y 2 \pi^2 (\cos^2(\pi y) - \sin^2(\pi y)) \sin^2(\pi x) + c(u_{\text{exact}})\]

The spatial derivatives are computed using second-order centered differences, with the data distributed over \(n_x \times n_y\) points on a uniform spatial grid. The problem is set up to use spatial grid parameters \(n_x=64\) and \(n_y=64\), with heat conductivity parameters \(k_x=1.0\) and \(k_y=1.0\).

This problem is solved via a fixed point iteration with Anderson acceleration, where the fixed point is function formed by adding \(u\) to both sides of the equation, i.e.,

\[b + u = \nabla \cdot (D \nabla u) + c(u) + u,\]

so that the fixed point function is

\[G(u) = \nabla \cdot (D \nabla u) + c(u) + u - b.\]

The problem is run using a tolerance of \(10^{-8}\), and a starting vector containing all ones. This example highlights the use of the various orthogonalization routine options within Anderson Acceleration, passed to the example problem via the --orthaa flag. Available options include 0 (KIN_ORTH_MGS), 1 (KIN_ORTH_ICWY), 2 (KIN_ORTH_CGS2), and 3 (KIN_ORTH_DCGS2).

The following tables contain all available input parameters when running the example problem.

Optional Input Parameter Flags

Flag	Description
--mesh <nx> <ny>	mesh points in the x and y directions
--np <npx> <npy>	number of MPI processes in the x and y directions
--domain <xu> <yu>	domain upper bound in the x and y direction
--k <kx> <ky>	diffusion coefficients
--rtol <rtol>	relative tolerance
--maa <maa>	number of previous residuals for Anderson Acceleration
--damping <damping>	damping parameter for Anderson Acceleration
--orthaa <orthaa>	orthogonalization routine used in Anderson Acceleration
--maxits <maxits>	max number of iterations
--c <cu>	nonlinear function choice (integer between 1 - 17)
--timing	print timing data
--help	print available input parameters and exit

Input Parameter Flags for Nonlinear Function \(c(u)\)

Flag	Function
--c 1	\(c(u) = u\)
--c 2	\(c(u) = u^3 - u\)
--c 3	\(c(u) = u - u^2\)
--c 4	\(c(u) = e^u\)
--c 5	\(c(u) = u^4\)
--c 6	\(c(u) = \cos^2(u) - \sin^2(u)\)
--c 7	\(c(u) = \cos^2(u) - \sin^2(u) - e^u\)
--c 8	\(c(u) = e^u u^4 - u e^{\cos(u)}\)
--c 9	\(c(u) = e^{(\cos^2(u))}\)
--c 10	\(c(u) = 10(u - u^2)\)
--c 11	\(c(u) = -13 + u + ((5-u)u - 2)u\)
--c 12	\(c(u) = \sqrt{5}(u - u^2)\)
--c 13	\(c(u) = (u - e^u)^2 + (u + u \sin(u) - \cos(u))^2\)
--c 14	\(c(u) = u + u e^u + u e^{-u}\)
--c 15	\(c(u) = u + u e^u + u e^{-u} + (u - e^u)^2\)
--c 16	\(c(u) = u + u e^u + u e^{-u} + (u - e^u)^2 + (u + u\sin(u) - \cos(u))^2\)
--c 17	\(c(u) = u + u e^{-u} + e^u (u + \sin(u) - \cos(u))^3\)

Parallel Example Using hypre: kin_heat2D_nonlin_hypre_pfmg

As an illustration of the use of the KINSOL package for the solution of nonlinear systems in parallel and using hypre linear solvers, we give a sample program called kin_heat2D_nonlin_hypre_pfmg.cpp. It uses the KINSOL fixed-point (KIN_FP) iteration with Anderson Acceleration and the MPI parallel vector for the solution of a steady-state 2D heat equation with an additional nonlinear term defined by \(c(u)\):

\[b = \nabla \cdot (D \nabla u) + c(u) \quad \text{in} \quad \mathcal{D} = [0,1] \times [0,1]\]

where \(D\) is a diagonal matrix with entries \(k_x\) and \(k_y\) for the diffusivity in the \(x\) and \(y\) directions, respectively. The boundary conditions are

\[u(0,y) = u(1,y) = u(x,0) = u(x,1) = 0.\]

We chose the analytical solution to be

\[u_{\text{exact}} = u(x,y) = \sin^2(\pi x) \sin^2(\pi y)\]

Hence, we define the static term \(b\) as follows

\[b = k_x 2 \pi^2 (\cos^2(\pi x) - \sin^2(\pi x)) \sin^2(\pi y) + k_y 2 \pi^2 (\cos^2(\pi y) - \sin^2(\pi y)) \sin^2(\pi x) + c(u_{\text{exact}})\]

The spatial derivatives are computed using second-order centered differences, with the data distributed over \(n_x \times n_y\) points on a uniform spatial grid. The problem is set up to use spatial grid parameters \(n_x=64\) and \(n_y=64\), with heat conductivity parameters \(k_x=1.0\) and \(k_y=1.0\).

This problem is solved via a fixed point iteration with Anderson acceleration, where the fixed point function is formed by implementing the Laplacian as a matrix-vector product,

\[b = A u + c(u)\]

and solving for \(u\) results in the fixed point function

\[G(u) = A^{-1} (b - c(u)).\]

The problem is run using a tolerance of \(10^{-8}\), and a starting vector containing all ones. The linear system solve is executed using the PCG linear solver with the hypre PFMG preconditioner. The setup of the linear solver can be found in the Setup_LS function, and setup of the hypre preconditioner can be found in the Setup_Hypre function. This example highlights the use of the various orthogonalization routine options within Anderson Acceleration, passed to the example problem via the --orthaa flag. Available options include 0 (KIN_ORTH_MGS), 1 (KIN_ORTH_ICWY), 2 (KIN_ORTH_CGS2), and 3 (KIN_ORTH_DCGS2).

All input parameter flags available for the previous example are also available for this problem. In addition, all runtime flags controlling the linear solver and hypre related parameters are set using the flags in the following table.

Optional Input Parameter Flags for hypre

Flag	Description
--lsinfo	output residual history for PCG
--liniters <liniters>	max number of iterations for PCG
--epslin <epslin>	linear tolerance for PCG
--pfmg_relax <pfmg_relax>	relaxation type in PFMG
--pfmg_nrelax <pfmg_nrelax>	pre/post relaxation sweeps in PFMG