A smoothness/shock indicator is proposed for the RKDG methods solving nonlinear conservation laws. A few numerical experiments are presented as evidence that the indicator helps in detecting shocks, high order discontinuities, regions of smooth solutions, and numerical “instability”. keywords. Conservation law, discontinuous Galerkin method, smoothness indicator. AMS subject class. Primary: 65M...

We study the parabolic approximation of a multidimensional scalar conservation law with initial and boundary conditions. We prove that the rate of convergence of the viscous approximation to the weak entropy solution is of order η, where η is the size of the artificial viscosity. We use a kinetic formulation and kinetic techniques for initial-boundary value problems developed by the last two au...

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pres...

In this comment we demonstrate that the example of a DC conducting wire of finite length as and example for the application of the Biot-Savart Law is flawed due to the non-conservation of electric charge for such an unphysical setting. The implications drawn in [1] on the restrictions of Ampère’s Circuital Law in integral form are unnecessary as long as only physically realizable situations obe...

2 to the Lorentz transformations via the law of addition of relativistic velocities. Abstract. In this paper we show how to get the Lorentz transformations from E = mc 2 , the laws of conservation of energy and momentum, and the special relativity principle. To this end we first deduce the law of addition of relativistic velocities. From E = mc 2 to the Lorentz transformations via the law of ad...

A fraction of solar active regions are observed to have current helicity of a sign that contradicts the polarity law for magnetic helicity; this law corresponds to the well-known Hale polarity law for sunspots. A significant excess of active regions with the ”wrong” sign of helicity is seen to occur just at the beginning of the cycle. We compare these observations with predictions from a dynamo...

Abstract : The Second Law of Thermodynamics asserts that the physical entropy of an adiabatic system is an increasing function in time. In this paper we will study a more stringent version of this law, according to which the entropy should not only increase in time, but the rate of increase is optimal in absolute value among all possible evolutions. We will establish this property in the framew...

Least-squares finite element methods (LSFEMs) for the inviscid Burgers equation are studied. The scalar nonlinear hyperbolic conservation law is reformulated by introducing the flux vector, or the associated flux potential, explicitly as additional dependent variables. This reformulation highlights the smoothness of the flux vector for weak solutions, namely f(u) ∈ H(div,Ω). The standard least-...

The Divergence Theorem as usually stated cannot be applied across a change of signature unless it is re-expressed to allow for a finite source term on the signature change surface. Consequently all conservation laws must also be ‘modified’, and therefore insistence on conservation of matter across such a surface cannot be physically justified. The Darmois junction conditions normally ensure con...

The traditional conservation of mass equation is derived using a first-order Taylor series to represent flux change in a control volume, which is valid strictly for cases of linear changes in flux through the control volume. We show that using higher-order Taylor series approximations for the mass flux results in mass conservation equations that are intractable. We then show that a fractional T...