Matrix components precompute these compact stencil coefficients and provides member functions that computes the finite difference approximation of input vectors. Unit scaled coefficients (assuming grid spacing is one) are computed and stored to be accessible via both CPU and GPU memory. On the GPU, the constant memory space is used for faster memory access~\cite{ch5:cudaguide}. In order to apply a stencil on a non unit-spaced grid, with grid space $\Delta x$, the scale factor $1/(\Delta x)^q$ will have to be multiplied by the finite difference sum, i.e., $(c_{00}u_0 + c_{01}u_1 + c_{02}u_2)/(\Delta x)^q \approx u^{(q)}_0$ as in the first row of \eqref{ch5:eq:stencilmatrix}.
Setting up a two-dimensional grid of size $N_x \times N_y$ in the unit square and computing the first derivative hereof is illustrated in Listing~\ref{ch5:lst:stencil}. The grid is a vector component, derived from the vector class. It is by default treated as a device object and memory is automatically allocated on the device to fit the grid size. The finite difference approximation as in \eqref{ch5:eq:fdstencil}, is performed via a CUDA kernel behind the scenes during the calls to \texttt{mult} and \texttt{diff\_x}, utilizing the memory hierarchy as the CUDA guidelines prescribe~\cite{ch5:cudaguide,ch5:cudapractice}. To increase developer productivity, kernel launch configurations have default settings, based on CUDA guidelines, principles, and experiences from performance testings, such that the user does not have to explicitly specify them. For problem-specific finite difference approximations, where the built-in stencil operators are insufficient, a pointer to the coefficient matrix \eqref{ch5:eq:stencilcoeffs} can be accessed as demonstrated in Listing \ref{ch5:lst:stencil} and passed to customized kernels.
-
+\pagebreak
\lstinputlisting[label=ch5:lst:stencil,caption={two-dimensional finite difference stencil example: computing the first derivative using five points ($\alpha=\beta=2$) per dimension, a total nine-point stencil}]{Chapters/chapter5/code/ex2.cu}
In the following sections we demonstrate how to go from an initial value problem (IVP) or a boundary value problem (BVP) to a working application solver by combining existing library components along with new custom-tailored components. We also demonstrate how to apply spatial and temporal domain decomposition strategies that can make existing solvers take advantage of systems equipped with multiple GPUs. Next section demonstrates how to rapidly assemble a PDE solver using library components.
\begin{align}\label{ch5:eq:discreteheateq}
\frac{\partial u}{\partial t} = \mymat{A}\myvec{u}, \qquad \mymat{A} \in \mathbb{R}^{N\times N}, \quad \myvec{u} \in \mathbb{R}^{N},
\end{align}
-where $\mymat{A}$ is the sparse finite difference matrix and $N$ is the number of unknowns in the discrete system. The temporal derivative is now free to be approximated by any suitable choice of a time-integration method\index{time integration}. The most simple integration scheme would be the first-order accurate explicit forward Euler method\index{forward Euler},
+where $\mymat{A}$ is the sparse finite difference matrix and $N$ is the number of unknowns in the discrete system. The temporal derivative is now free to be approximated by any suitable choice of a time-integration method\index{time integration}. The most simple integration scheme would be the first-order accurate explicit forward Euler method\index{Euler!forward Euler},
\begin{align}\label{ch5:eq:forwardeuler}
\myvec{u}^{n+1} = \myvec{u}^n + \delta t\,\mymat{A}\myvec{u}^n,
\end{align}
\begin{align}
\mymat{A}\myvec{u}=\myvec{f}, \qquad \myvec{u},\myvec{f} \in \mathbb{R}^{N}, \quad \mymat{A} \in \mathbb{R}^{N\times N}, \label{ch5:eq:poissonsystem}
\end{align}
-where $\mymat{A}$ is the sparse matrix formed by finite difference coefficients, $N$ is the number of unknowns, and $\myvec{f}$ is given by \eqref{ch5:eq:poissonrhs}. Equation \eqref{ch5:eq:poissonsystem} can be solved in numerous ways, but a few observations may help do it more efficiently. Direct solvers based on Gaussian elimination are accurate and use a finite number of operations for a constant problem size. However, the arithmetic complexity grows with the problem size by as much as $\mathcal{O}(N^3)$ and does not exploit the sparsity of $\mymat{A}$. Direct solvers are therefore mostly feasible for dense systems of limited sizes. Sparse direct solvers exist, but they are often difficult to parallelize, or applicable for only certain types of matrices. Regardless of the discretization technique, the discretization of an elliptic PDE into a linear system as in \eqref{ch5:eq:poissonsystem} yields a very sparse matrix $\mymat{A}$ when $N$ is large. Iterative methods\index{iterative methods} for solving large sparse linear systems find broad use in scientific applications, because they require only an implementation of the matrix-vector product, and they often use a limited amount of additional memory. Comprehensive introductions to iterative methods may be found in any of~\cite{ch5:Saad2003,ch5:Kelley1995,ch5:Barrett1994}.
+where $\mymat{A}$ is the sparse matrix formed by finite difference coefficients, $N$ is the number of unknowns, and $\myvec{f}$ is given by \eqref{ch5:eq:poissonrhs}. Equation \eqref{ch5:eq:poissonsystem} can be solved in numerous ways, but a few observations may help do it more efficiently. Direct solvers based on Gaussian elimination are accurate and use a finite number of operations for a constant problem size. However, the arithmetic complexity grows with the problem size by as much as $\mathcal{O}(N^3)$ and does not exploit the sparsity of $\mymat{A}$. Direct solvers are therefore mostly feasible for dense systems of limited sizes. Sparse direct solvers exist, but they are often difficult to parallelize, or applicable for only certain types of matrices. Regardless of the discretization technique, the discretization of an elliptic PDE into a linear system as in \eqref{ch5:eq:poissonsystem} yields a very sparse matrix $\mymat{A}$ when $N$ is large. Iterative methods\index{iterative method} for solving large sparse linear systems find broad use in scientific applications, because they require only an implementation of the matrix-vector product, and they often use a limited amount of additional memory. Comprehensive introductions to iterative methods may be found in any of~\cite{ch5:Saad2003,ch5:Kelley1995,ch5:Barrett1994}.
One benefit of the high abstraction level and the template-based library design is to allow developers to implement their own components, such as iterative methods for solving sparse linear systems. The library includes three popular iterative methods: conjugate gradient, defect correction\index{defect correction}, and geometric multigrid. The conjugate gradient method is applicable only to systems with symmetric positive definite matrices. This is true for the two-dimensional Poisson problem, when it is discretized with a five-point finite difference stencil, because then there will be no off-centered approximations near the boundary. For high-order approximations ($\alpha>1$), we use the defect correction method with multigrid preconditioning. See e.g., \cite{ch5:Trottenberg2001} for details on multigrid methods.
