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Stencil codes 196
Stencil codes
Stencil codes are a class of iterative kernels [1] which update array
elements according to some fixed pattern, called stencil [2] . They are
most commonly found in the codes of computer simulations, e.g. for
computational fluid dynamics in the context of scientific and
engineering applications. Other notable examples include solving
partial differential equations [1] , the Jacobi kernel, the Gauss–Seidel
method [2] , image processing [1] and cellular automata. [3] The regular
structure of the arrays sets stencil codes apart from other modeling
methods such as the Finite element method. Most finite difference
codes which operate on regular grids can be formulated as stencil
codes.
Definition
Stencil codes perform a sequence of sweeps (called timesteps) through
a given array. [2] Generally this is a 2- or 3-dimensional regular grid. [3]
The shape of a 6-point 3D von Neumann style
The elements of the arrays are often referred to as cells. In each timestep, the stencil code updates all array
elements. [2] Using neighboring array elements in a fixed pattern (called the stencil), each cell's new value is
computed. In most cases boundary values are left unchanged, but in some cases (e.g. LBM codes) those need to be
adjusted during the course of the computation as well. Since the stencil is the same for each element, the pattern of
data accesses is repeated. [4]
stencil.
More formally, we may define stencil codes as a 5-tuple with the following meaning: [3]
• is the index set. It defines the topology of the array.
• is the (not necessarily finite) set of states, one of which each cell may take on on any given timestep.
• defines the initial state of the system at time 0.
• is the stencil itself and describes the actual shape of the neighborhood. (There are elements in the
stencil.
• is the transition function which is used to determine a cell's new state, depending on its neighbors.
Since I is a k-dimentional integer interval, the array will always have the topology of a finite regular grid. The array
is also called simulation space and individual cells are identified by their index . The stencil is an ordered set
of relative coordinates. We can now obtain for each cell the tuple of its neighbors indices
Their states are given by mapping the tuple to the corresponding tuple of states :
This is all we need to define the system's state for the following time steps with :
Note that is defined on and not just on since the boundary conditions need to be set, too. Sometimes the
elements of may be defined by a vector addition modulo the simulation space's dimension to realize toroidal