Supra-convergent schemes on irregular grids

Abstract

As Tikhonov and Samarskiĭ showed for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is not essential that <italic>k</italic>th-order compact difference schemes be centered at the arithmetic mean of the stencil’s points to yield second-order convergence (although it does suffice). For stable schemes and even <italic>k</italic>, the main point is seen when the <italic>k</italic>th difference quotient is set equal to the value of the <italic>k</italic>th derivative at the middle point of the stencil; the proof is particularly transparent for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For any <italic>k</italic>, in fact, there is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left floor k slash 2 right floor"> <mml:semantics> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo>⌋</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \lfloor {k/2} \right \rfloor</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-parameter family of symmetric averages of the values of the <italic>k</italic>th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov’s tridiagonal scheme (approximating <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D squared y equals f"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{D^2}y = f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with third-order truncation error) yields fourth-order convergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three-periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of <italic>k</italic> variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.