A fourth-order accurate finite-volume method is presented for solving timedependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space are combined with detailed mechanisms for accommodating the adapting grids. These considerations ensure that conservation is maintained and that the divergence of a constant vector field is always zero (freestream-preservation property). The solution in time is advanced with a fourth-order Runge-Kutta method. A series of tests verifies that the expected accuracy is achieved in smooth flows and the solution of a Mach reflection problem demonstrates the effectiveness of the algorithm in resolving strong discontinuities.

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SUMMARYA fourth-order finite-volume method for solving the Navier-Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth-order quadrature rules for evaluating face-averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge-Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth-order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non-rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries.

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