Well-posedness of one-way wave equations and absorbing boundary conditions

Abstract

A one-way wave equation is a partial differential equation which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. Such equations are used in geophysics, in underwater acoustics, and as numerical "absorbing boundary conditions". Their construction can be reduced to the approximation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 1 minus s squared EndRoot"> <mml:semantics> <mml:msqrt> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {1 - {s^2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 1 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[ - 1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by a rational function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis s right-parenthesis equals p Subscript m Baseline left-parenthesis s right-parenthesis slash q Subscript n Baseline left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r(s) = {p_m}(s)/{q_n}(s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This paper characterizes those rational functions <italic>r</italic> for which the corresponding one-way wave equation is well posed, both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis s right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r(s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> interpolates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot 1 minus s squared EndRoot"> <mml:semantics> <mml:msqrt> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {1 - {s^2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at sufficiently many points in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative 1 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">( - 1,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then well-posedness is assured. It follows that absorbing boundary conditions based on Padé approximation are well posed if and only if (<italic>m, n</italic>) lies in one of two distinct diagonals in the Padé table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived by Chebyshev or least-squares approximation.