A new very high-order technique for solving conservation laws with curved boundary domains is proposed. A Finite Difference scheme on Cartesian grids is coupled with an original ghost cell method that provide accurate approximations for smooth solutions. The technology is based on a specific least square method with restrictions that enables to handle general Robin conditions. Several examples in two-dimensional geometries are presented for the unsteady Convection–Diffusion equation and the Euler equations. A fifth-order WENO scheme is employed with matching fifth-order reconstruction at the boundaries. Arbitrary high-order reconstruction for smooth flows is achievable independently of the underlying differential equation since the method works as a black-box dedicated to boundary condition treatment.This work has been partially supported by the Ministerio de Economı́a y Competitividad (grant #DPI2015-
68431-R) and #RTI2018-093366-B-I00 of the Ministerio de Ciencia, Innovación y Universidades of the Spanish
Government and by the Consellerı́a de Educación e Ordenación Universitaria of the Xunta de Galicia (grants
#GRC2014/039 and #ED431C 2018/41), cofinanced with FEDER, Spain funds and the Universidade da Coruña, Spain. J. Fernandez-Fidalgo gratefully acknowledges the contributions of the IACOBUS Program, Spain and the INDITEX-UDC, Spain grant that have partially financed the present work. S. Clain acknowledges the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, Portugal, through COMPETE 2020 – Programa Operational Fatores de Competitividade, and the National Funds through FCT — Fundação para a Ciência e a Tecnologia, Portugal, project No. UID/FIS/04650/2013 and project No. POCI-01-0145-FEDER-02811
