New adaptive artificial viscosity method for hyperbolic systems of conservation laws

We propose a new finite volume method for solving general multidimensional hyperbolic systems of conservation laws. Our method is based on an appropriate numerical flux and a high-order piecewise polynomial reconstruction. The latter is utilized without any computationally expensive nonlinear limiters, which are typically needed to guarantee nonlinear stability of the scheme. Instead, we enforce stability of the proposed method by adding a new adaptive artificial viscosity, whose coefficients are proportional to the size of the weak local residual, which is sufficiently large (∼ Δ , where Δ is a discrete small scale) at the shock regions, much smaller (∼ Δ α , where α is close to 2) near the contact waves, and very small (∼ Δ 4 ) in the smooth parts of the computed solution. t the proposed scheme on a number of benchmarks for both scalar conservation laws and for one- and two-dimensional Euler equations of gas dynamics. The obtained numerical results clearly demonstrate the robustness and high accuracy of the new method.