A Residual-Based Numerical Viscosity Regularization Approach for Higher-order Finite Volume Discretization of Scalar Hyperbolic Conservation Laws

A Space-time finite volume method is utilized to construct a parameterized family of two-step explicit higher-order schemes for scalar hyperbolic conservation laws. Utilizing a local space-time expansion of the flux-integral form of the equation error, generalized quadratures of local grid functions of the solution and the associated local source term are formulated to couple grid points within the domains of dependence and influence of new updates about the centroid of each space-time control volume. Optimal quadrature parameters for the discretization are then determined through a minimization of the error expansion to account for local space-time fluxes to all neighboring mesh points within the computational domain. % Hence, a more accurate space-time descriptions of the leading numerical viscosity coefficients in the residual errors are then characterized based on the space-time coupling of the desired set of mesh points utilized in the discretization about the centroid. Consequently, the quadrature weights and the time step sizes are optimized to control and regularize the residual errors to minimize nonphysical oscillations. Numerical experiments demonstrate the effectiveness of the discretization method in minimizing the associated nonphysical oscillations in numerical solutions.