In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner.~We first derive continuous energy estimates,~and then proceed to the discrete setting.~We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method.~By applying the discrete energy method and imitating the continuous analysis,~the discrete estimate that resembles the continuous counterpart is obtained proving stability.~We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator.~Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented.~The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier Stokes equations.

翻译：本文提出一种高阶精确方法，以可证明稳定的方式求解不可压缩边界层方程。我们首先推导连续能量估计，进而转向离散设定。采用求和分部形式的高阶有限差分法构建离散近似，并通过同时逼近项方法弱实现边界条件。通过应用离散能量法并模仿连续分析，获得与连续估计相似的离散估计，从而证明稳定性。此外，我们证明新推导的边界条件消除了非线性离散空间算符零空间相关的奇异性。给出验证该格式高阶精度且与理论结果一致的数值实验。数值结果与著名的Blasius相似解以及不可压缩Navier-Stokes方程的解进行了比较。