Abstract
In this article we develop a Primitive Variable Recovery Scheme (PVRS) to solve any system of coupled differential conservative equations. This method obtains directly the primitive variables applying the chain rule to the time term of the conservative equations. With this, a traditional finite volume method for the flux is applied in order avoid violation of both, the entropy and “Rankine-Hugoniot” jump conditions. The time evolution is then computed using a forward finite difference scheme. This numerical technique evades the recovery of the primitive vector by solving an algebraic system of equations as it is often used and so, it generalises standard techniques to solve these kind of coupled systems. The article is presented bearing in mind special relativistic hydrodynamic numerical schemes with an added pedagogical view in the appendix section in order to easily comprehend the PVRS. We present the convergence of the method for standard shock-tube problems of special relativistic hydrodynamics and a graphical visualisation of the errors using the fluctuations of the numerical values with respect to exact analytic solutions. The PVRS circumvents the sometimes arduous computation that arises from standard numerical methods techniques, which obtain the desired primitive vector solution through an algebraic polynomial of the charges.
Highlights
The use of numerical methods to solve differential equations has constituted a substantial amount of work since the conception of approximate solutions to a given set of equations
The method developed is general and valid to any set of coupled conservative equations. We show how this method can be applied in the particular case of 1D special relativistic hydrodynamics (1DRHD)
In this article we have developed a new numerical algorithm to solve any set of coupled differential conservative equations for which the primitive variable vector u is directly obtained
Summary
The use of numerical methods to solve differential equations has constituted a substantial amount of work since the conception of approximate solutions to a given set of equations. In this article we show how it is possible to construct a general numerical iteration method, using a combination of finite differences and finite volume integration techniques for the time and spatial evolutions respectively, to directly find the solutions u avoiding any middle cumbersome step such as the ones mentioned above. In order to compare the numerical solution with experiments and/or observations, a set of primitive physical measurable variables u needs to be constructed For this particular case, this primitive variable set is given by by the pressure p, the velocity along three spacial dimensions vi and, the particle number density n. In order to avoid this cumbersome task, we show how it is possible to obtain a direct numerical solution of the primitive variables, which is valid for all conservative equation systems (cf. Eq (15))
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