In the present work we derive and study a non-linear elliptic PDE coming from the problem of estimation of sound speed inside the Earth. The physical setting of the PDE allows us to pose only a Cauchy problem, and hence is ill-posed. However, we are still able to solve it numerically on a long enough time interval to be of practical use. We used two approaches. The first approach is a finite di...

We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov’s method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is inv...

Numerical simulations often provide strong evidences for the existence and stability of discrete shocks for certain nite diierence schemes approximating conservation laws. This paper presents a framework for converting such numerical observations to mathematical proofs. The framework is applicable to conservative schemes approximating stationary shocks of one dimensional scalar conservation law...

Gasdynamic features of slowly rotating gas axisymmetric accretion onto a gravitating center are investigated. The process of flow restructuring is studied as the angular velocity of accreting matter approaches the Keplerian angular velocity. For spherically-symmetric accretion, the conditions are found of the existence of a steady-state supersonic solution for various external boundary conditio...

We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose a fifth-order Weighted PowerENO sweeping scheme...

We introduce two kinds of explicit solutions to the convection-reaction equation, ut + (|u| /q)x = u, u, x ∈ R, t ∈ R , q > 1, and employ them to test properties of various computational schemes. From this test we observe that computed solutions using Lax-Friedrichs, MacCormack and Lax-Wendroff schemes break down in a finite time. On the other hand some other schemes including WENO, NT and Godu...

In classical continuum mechanics, quasi-linear systems of conservation laws can be symmetrized if they admit an additional convex conservation law. In particular, this implies the hyperbolicity of governing equations. For capillary fluids, the internal energy depends not only on the density but also on its derivatives with respect to space variables. Consequently, the governing equations belong...