Hybrid Fourier-Continuation Method and Weighted Essentially Non-oscillatory Finite Difference Scheme for Hyperbolic Conservation Laws in a Single-Domain Framework

We investigate a hybrid Fourier-Continuation (FC) method (Bruno and Lyon, J Comput Phys 229:2009---2033, 2010) and fifth order characteristic-wise weighted essentially non-oscillatory (WENO) finite difference scheme for solving system of hyperbolic conservation laws on a uniformly discretized Cartesian domain. The smoothness of the solution is measured by the high order multi-resolution algorithm by Harten (J Comput Phys 49:357---393, 1983) at each grid point in a single-domain framework (Costa and Don, J Comput Appl Math 204(2):209---218, 2007) (Hybrid), as opposed to each subdomain in a multi-domain framework (Costa et al., J Comput Phys 224(2):970---991, 2007; Shahbazi et al., J Comput Phys 230:8779---8796, 2011). The Hybrid scheme conjugates a high order shock-capturing WENO-Z5 (nonlinear) scheme (Borges et al., J Comput Phys 227:3101---3211, 2008) in non-smooth WENO stencils with an essentially non-dissipative and non-dispersive FC (linear) method in smooth FC stencils, yielding a high fidelity scheme for applications containing both discontinuous and complex smooth structures. Several critical and unique numerical issues in an accurate and efficient implementation (such as reasonable choice of parameters, singular value decomposition, fast Fourier transform, symmetry preservation, and overlap zone) of the FC method, due to a dynamic spatial and temporal change in the size of data length in smooth FC stencils in a single-domain framework, will be illustrated and addressed. The accuracy and efficiency of the Hybrid scheme in solving one and two dimensional system of hyperbolic conservation laws is demonstrated with several classical examples of shocked flow, such as the one dimensional Riemann initial value problems (123, Sod and Lax), the Mach 3 shock---entropy wave interaction problem with a small entropy sinusoidal perturbation, the Mach 3 shock---density wave interaction problem, and the two dimensional Mach 10 double Mach reflection problem. For a sufficiently large problem size, a factor of almost two has been observed in the speedup of the Hybrid scheme over the WENO-Z5 scheme.