Rates of convergence for viscous splitting of the Navier-Stokes equations

Abstract

Viscous splitting algorithms are the underlying design principle for many numerical algorithms which solve the Navier-Stokes equations at high Reynolds number. In this work, error estimates for splitting algorithms are developed which are uniform in the viscosity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi>ν</mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as it becomes small for either two- or three-dimensional fluid flow in all of space. In particular, it is proved that standard viscous splitting converges uniformly at the rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C nu normal upper Delta t"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>ν</mml:mi> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C\nu \Delta t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, Strang-type splitting converges at the rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C nu left-parenthesis normal upper Delta t right-parenthesis squared"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>ν</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">C\nu {(\Delta t)^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and also that solutions of the Navier-Stokes and Euler equations differ by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C nu"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in this case. Here <italic>C</italic> depends only on the time interval and the smoothness of the initial data. The subtlety in the analysis occurs in proving these estimates for fixed large time intervals for solutions of the Navier-Stokes equations in two space dimensions. The authors derive a new long-time estimate for the two-dimensional Navier-Stokes equations to achieve this. The results in three space dimensions are valid for appropriate short time intervals; this is consistent with the existing mathematical theory.