Abstract A general strategy exists for constructing Energy Stable Weighted Essentially Non–Oscillatory (ESWENO) ﬁnite diﬀerence schemes up to eighth-order on periodic do-mains. These ESWENO schemes satisfy an energy norm stability proof for bothcontinuous and discontinuous solutions of systems of linear hyperbolic equations.Herein, boundary closures are developed for the fourth-order ESWENO scheme thatmaintain wherever possible the WENO stencil biasing properties, while satisfyingthe summation-by-parts (SBP) operator convention, thereby ensuring stability inan L 2 norm. Second-order, and third-order boundary closures are developed thatachieve stability in diagonal and block norms, respectively. The global accuracyfor the second-order closures is three, and for the third-order closures is four. Anovel set of non-uniform ﬂux interpolation points is necessary near the boundariesto simultaneously achieve 1) accuracy, 2) the SBP convention, and 3) WENO stencilbiasing mechanics. Contents 1 Introduction 22 Deﬁnitions 42.1 Complementary Grids . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Summation-By-Parts Operators . . . . . . . . . . . . . . . . . . . . . 52.2.1 Central Fourth-order SBP Operators . . . . . . . . . . . . . . 63 Expressing Summation-By-Parts Operators in Flux Form 83.1 General Structure of ESWENO Schemes . . . . . . . . . . . . . . . . 83.2 Nonuniform Flux-Points . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Interpolation Operators and Preferred Weights . . . . . . . . . . . . 114 Energy Stabilization Terms 115 Numerical Tests 125.1 Eigenvalues of the ESWENO operator . . . . . . . . . . . . . . . . . 135.2 Linear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2.1 Sine Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2.2 Square Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3.1 Subsonic One-Dimensional Nozzle . . . . . . . . . . . . . . . 175.3.2 Transonic One-Dimensional Nozzle . . . . . . . . . . . . . . . 175.3.3 Two Dimensional Inviscid Vortex . . . . . . . . . . . . . . . . 185.4 Supersonic Shear Layers . . . . . . . . . . . . . . . . . . . . . . . . . 196 Conclusions 21A Smoothness Indicators 231
['Travis C. Fisher', 'Mark H. Carpenter', 'Nail K. Yamaleev', 'Steven H. Frankel']
Boundary Closures for Fourth-order Energy Stable Weighted Essentially Non-Oscillatory Finite Difference Schemes