636 research outputs found
The convergence rate of approximate solutions for nonlinear scalar conservation laws
The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates
On the inadmissibility of non-evolutionary shocks
In recent years, numerical solutions of the equations of compressible magnetohydrodynamic (MHD) flows have been found to contain intermediate shocks for certain kinds of problems. Since these results would seem to be in conflict with the classical theory of MHD shocks, they have stimulated attempts to reexamine various aspects of this theory, in particular the role of dissipation. In this paper, we study the general relationship between the evolutionary conditions for discontinuous solutions of the dissipation-free system and the existence and uniqueness of steady dissipative shock structures for systems of quasilinear conservation laws with a concave entropy function. Our results confirm the classical theory. We also show that the appearance of intermediate shocks in numerical simulations can be understood in terms of the properties of the equations of planar MHD, for which some of these shocks turn out to be evolutionary. Finally, we discuss ways in which numerical schemes can be modified in order to avoid the appearance of intermediate shocks in simulations with such symmetry
A multidimensional grid-adaptive relativistic magnetofluid code
A robust second order, shock-capturing numerical scheme for multi-dimensional
special relativistic magnetohydrodynamics on computational domains with
adaptive mesh refinement is presented. The base solver is a total variation
diminishing Lax-Friedrichs scheme in a finite volume setting and is combined
with a diffusive approach for controlling magnetic monopole errors. The
consistency between the primitive and conservative variables is ensured at all
limited reconstructions and the spatial part of the four velocity is used as a
primitive variable. Demonstrative relativistic examples are shown to validate
the implementation. We recover known exact solutions to relativistic MHD
Riemann problems, and simulate the shock-dominated long term evolution of
Lorentz factor 7 vortical flows distorting magnetic island chains.Comment: accepted for publication in Computer Physics Communication
A Constrained Transport Method for the Solution of the Resistive Relativistic MHD Equations
We describe a novel Godunov-type numerical method for solving the equations
of resistive relativistic magnetohydrodynamics. In the proposed approach, the
spatial components of both magnetic and electric fields are located at zone
interfaces and are evolved using the constrained transport formalism. Direct
application of Stokes' theorem to Faraday's and Ampere's laws ensures that the
resulting discretization is divergence-free for the magnetic field and
charge-conserving for the electric field. Hydrodynamic variables retain,
instead, the usual zone-centred representation commonly adopted in
finite-volume schemes. Temporal discretization is based on Runge-Kutta
implicit-explicit (IMEX) schemes in order to resolve the temporal scale
disparity introduced by the stiff source term in Ampere's law. The implicit
step is accomplished by means of an improved and more efficient Newton-Broyden
multidimensional root-finding algorithm. The explicit step relies on a
multidimensional Riemann solver to compute the line-averaged electric and
magnetic fields at zone edges and it employs a one-dimensional Riemann solver
at zone interfaces to update zone-centred hydrodynamic quantities. For the
latter, we introduce a five-wave solver based on the frozen limit of the
relaxation system whereby the solution to the Riemann problem can be decomposed
into an outer Maxwell solver and an inner hydrodynamic solver. A number of
numerical benchmarks demonstrate that our method is superior in stability and
robustness to the more popular charge-conserving divergence cleaning approach
where both primary electric and magnetic fields are zone-centered. In addition,
the employment of a less diffusive Riemann solver noticeably improves the
accuracy of the computations.Comment: 25 pages, 14 figure
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
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