blitz_4.html

A C++ class library for scientific computing

These curl operators return three-dimensional <CODE>TinyVector</CODE>s of the 
appropriate numeric type:
</P><P>

<DL COMPACT>
<DT><CODE>curl(Vx,Vy,Vz)</CODE>
<DD>2nd order curl operator using the central12 operator.  Factor: <EM>2h</EM>
<P>

<DT><CODE>curl4(Vx,Vy,Vz)</CODE>
<DD>4th order curl operator using the central14 operator.  Factor: <EM>12h</EM>
<P>

<DT><CODE>curl(V)</CODE>
<DD>2nd order curl operator on a 3D vector field (e.g.
<CODE>Array&#60;TinyVector&#60;double,3&#62;,3&#62;</CODE>, using the central12 operator.  Factor:
<EM>2h</EM>
<P>

<DT><CODE>curl4(V)</CODE>
<DD>4th order curl operator on a 3D vector field, using the central14 operator.
Factor: <EM>12h</EM>
</DL>
<P>

Note that the above are available in normalized versions <CODE>curln</CODE> and
<CODE>curl4n</CODE>, which have factors of <CODE>h</CODE>.
</P><P>

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<H3> 4.4.9 Divergence (<EM>nabla cdot</EM>) operators </H3>
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<P>

The divergence operators return a scalar value.
</P><P>

<DL COMPACT>
<DT><CODE>div(Vx,Vy)</CODE>
<DD>2nd order div operator using the central12 operator.  Factor: <EM>2h</EM>
<P>

<DT><CODE>div4(Vx,Vy)</CODE>
<DD>4th order div operator using the central14 operator.  Factor: <EM>12h</EM>
<P>

<DT><CODE>div(Vx,Vy,Vz)</CODE>
<DD>2nd order div operator using the central12 operator.  Factor: <EM>2h</EM>
<P>

<DT><CODE>div4(Vx,Vy,Vz)</CODE>
<DD>4th order div operator using the central14 operator.  Factor: <EM>12h</EM>
<P>

<DT><CODE>div2D(V)</CODE>
<DD>2nd order div operator on a 2D vector field, using the central12 operator.
Factor: <EM>2h</EM>
<P>

<DT><CODE>div2D4(V)</CODE>
<DD>2nd order div operator on a 2D vector field, using the central14 operator.
Factor: <EM>12h</EM>
<P>

<DT><CODE>div3D(V)</CODE>
<DD>2nd order div operator on a 3D vector field, using the central12 operator.
Factor: <EM>2h</EM>
<P>

<DT><CODE>div3D4(V)</CODE>
<DD>2nd order div operator on a 3D vector field using the central14 operator.
Factor: <EM>12h</EM>
</DL>
<P>

These are available in normalized versions
<CODE>divn</CODE>, <CODE>div4n</CODE>, <CODE>div2Dn</CODE>, <CODE>div2D4n</CODE>, <CODE>div3Dn</CODE>, and
<CODE>div3D4n</CODE> which have factors of <EM>h</EM>.
</P><P>

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<H3> 4.4.10 Mixed partial derivatives </H3>
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<P>

<DL COMPACT>
<DT><CODE>mixed22(A,dim1,dim2)</CODE>
<DD>2nd order accurate, 2nd mixed partial derivative.  Factor: <EM>4h^2</EM>
<P>

<DT><CODE>mixed24(A,dim1,dim2)</CODE>
<DD>4th order accurate, 2nd mixed partial derivative.  Factor: <EM>144h^2</EM>
</DL>
<P>

There are also normalized versions of the above, <CODE>mixed22n</CODE> and
<CODE>mixed24n</CODE> which have factors <EM>h^2</EM>.
</P><P>

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<H2> 4.5 Declaring your own stencil operators </H2>
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<P>

You can declare your own stencil operators using the macro
<CODE>BZ_DECLARE_STENCIL_OPERATOR1</CODE>.  For example, here is the declaration
of <CODE>Laplacian2D</CODE>:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>BZ_DECLARE_STENCIL_OPERATOR1(Laplacian2D, A)
    return -4*A(0,0) + A(-1,0) + A(1,0) + A(0,-1) + A(0,1);
BZ_END_STENCIL_OPERATOR
</pre></td></tr></table></P><P>

To declare a stencil operator on 3 operands, use the macro
<CODE>BZ_DECLARE_STENCIL_OPERATOR3</CODE>.  Here is the declaration of <CODE>div</CODE>:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>BZ_DECLARE_STENCIL_OPERATOR3(div,vx,vy,vz)
  return central12(vx,firstDim) + central12(vy,secondDim)
    + central12(vz,thirdDim);
BZ_END_STENCIL_OPERATOR
</pre></td></tr></table></P><P>

The macros aren't magical; they just declare an inline template function
with the names and arguments you specify.  For example, the declaration of
<CODE>div</CODE> could also be written
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>template&#60;class T&#62;                              
inline typename T::T_numtype div(T&#38; vx, T&#38; vy, T&#38; vz)   
{
  return central12(vx,firstDim) + central12(vy,secondDim)
                                + central12(vz,thirdDim);
}
</pre></td></tr></table></P><P>

The template parameter <CODE>T</CODE> is an iterator type for arrays.
</P><P>

You are encouraged to use the macros when possible, because it is possible
the implementation could be changed in the future.
</P><P>

To declare a difference operator, use this syntax:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>BZ_DECLARE_DIFF(central12,A) {
  return A.shift(1,dim) - A.shift(-1,dim);
}
</pre></td></tr></table></P><P>

The method <CODE>shift(offset,dim)</CODE> retrieves the element at
<CODE>offset</CODE> in dimension <CODE>dim</CODE>.
</P><P>

Stencil operator declarations cannot occur inside a function.  If
declared inside a class, they are scoped by the class.
</P><P>

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<H2> 4.6 Applying a stencil </H2>
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<P>

The syntax for applying a stencil is:
</P><P>

<TABLE><tr><td>&nbsp;</td><td class=example><pre>applyStencil(stencilname(),A,B,C...,F);
</pre></td></tr></table></P><P>

Where <CODE>stencilname</CODE> is the name of the stencil, and <CODE>A,B,C,...,F</CODE>
are the arrays on which the stencil operates.
</P><P>

For examples, see <TT>`examples/stencil.cpp'</TT> and <TT>`examples/stencil2.cpp'</TT>.
</P><P>

Blitz++ interrogates the stencil object to find out how large its footprint
is.  It only applies the stencil over the region of the arrays where it
won't overrun the boundaries.
</P><P>

<A NAME="Customised Arrays"></A>
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