Stability and error analysis of a new class of higher-order consistent splitting schemes for the Navier-Stokes equations

A new class of fully decoupled consistent splitting schemes for the Navier-Stokes equations are constructed and analyzed in this paper. The schemes are based on the Taylor expansion at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Superscript n plus beta"> <mml:semantics> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">t^{n+\beta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta \ge 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> being a free parameter. It is shown that by choosing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 3 comma 6 comma 9"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta = 3, \,6,\,9</mml:annotation> </mml:semantics> </mml:math> </inline-formula> respectively for the second-, third- and fourth-order schemes, their numerical solutions are uniformed bounded in a strong norm, and admit optimal global-in-time convergence rates in both 2D and 3D. These results are the first stability and convergence results for any fully decoupled, higher than second-order schemes for the Navier-Stokes equations. Numerical results are provided to show that the third- and fourth-order schemes based on the usual backward differentiation formula (BDF) (i.e. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 1"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta =1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) are not unconditionally stable while the new third- and fourth-order schemes with suitable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi>β</mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are unconditionally stable and lead to expected convergence rates.