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Exact Solution of the 1D Riemann Problem in Newtonian and Relativistic Hydrodynamics

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Exact Solution of the 1D Riemann Problem in Newtonian and Relativistic Hydrodynamics EDUCATION Revista Mexicana de F´ısica E 59 (2013) 28–50 JANUARY–JUNE 2013 Exact solution of the 1D riemann problem in Newtonian and relativistic hydrodynamics F. D. Lora-Clavijo, J. P. Cruz-Perez,´ F. Siddhartha Guzman,´ and J. A. Gonzalez´ Instituto de F´ısica y Matematicas,´ Universidad Michoacana de San Nicolas´ de Hidalgo, Edificio C-3, Cd. Universitaria, 58040 Morelia, Michoacan,´ Mexico.´ Received 10 August 2012; accepted 15 February 2013 Some of the most interesting scenarios that can be studied in astrophysics, contain fluids and plasma moving under the influence of strong gravitational fields. To study these problems it is required to implement numerical algorithms robust enough to deal with the equations describing such scenarios, which usually involve hydrodynamical shocks. It is traditional that the first problem a student willing to develop research in this area is to numerically solve the one dimensional Riemann problem, both Newtonian and relativistic. Even a more basic requirement is the construction of the exact solution to this problem in order to verify that the numerical implementations are correct. We describe in this paper the construction of the exact solution and a detailed procedure of its implementation. Keywords: Hydrodynamics-astrophysical applications; hydrodynamics-fluids. PACS: 95.30.Lz; 47.35.-i 1. Introduction mentation of the solution. We focus on the solution of the problem and omit some of the mathematical background that High energy astrophysics has become one of the most impor- is actually very well described in the literature. tant subjects in astrophysics because it involves phenomena The paper is organized as follows. In Sec. 2 we present associated to high energy radiation, modeled with sources the Newtonian Riemann problem and how to implement it; traveling at high speeds or sources under the influence of in Sec. 2 we present the exact solution to the relativistic case strong gravitational fields like those due to black holes or and how to implement it. Finally in Sec. 3 we present some compact stars. Current models involve a hydrodynamical de- final comments. scription of the luminous source, and therefore hydrodynam- ical equations have to be solved. In this scenario, due to the complexity of the system of 2. Riemann problem for the Newtonian Euler equations it is required to apply numerical methods able to equations control the physical discontinuities arising during the evolu- tion of initial configurations, for example the evolution of the The Riemann problem is an initial value problem for a gas front shock in a supernova explosion, the front shock of a jet with discontinuous initial data, whose evolution is ruled by propagating in space, the edges of an accretion disk, or any Euler’s equations. The set of Euler’s equations determine the shock formed during a violent process. The study of these evolution of the density of gas, its velocity field and either its systems involve the implementation of advanced numerical pressure or total energy. A comfortable way of writing such methods, being two of the most efficient and robust ones the equations involves a flux balance form as follows high resolution shock capturing methods and smooth particle hydrodynamics which are representative of Eulerian and La- @tu + @xF(u) = 0 (1) grangian descriptions of hydrodynamics, each one with pros T T and cons. where u = (u1; u2; u3) = (½; ½v; E) is a set of conser- It is traditional that a first step to evaluate how appropri- vative variables and F is a flux vector, where ½ is the mass ate the implementation of a numerical method is, requires the density of the gas, v its velocity and E = ½((1=2)v2 + "), comparison of numerical results with an exact solution in a with " the specific internal energy of the gas. The enthalpy simple situation. The simplest problem in hydrodynamics is of the system is given by the expression H = (1=2)v2 + h, the 1D Riemann problem. This is an excellent test case be- where h is the specific internal enthalpy given by h = "+p=½, cause it has an exact solution in the Newtonian case (e.g. [1]) where p is the pressure of the gas. The fluxes are explicitly and also in the relativistic regime [2,3], where codes deal- in terms of the primitive variables ½; v; p and the conservative ing with high Lorentz factors are expected to work properly. variables [1] From our experience we have found that the existent liter- 0 1 0 1 ature about the construction of the exact solution is not as u2 ½v 2 2 B 1 u2 C explicit as it may be expected by students having their first @ A (3 ¡ ¡) + (¡ ¡ 1)u3 F(u)= ½v + p = @ 2 u1 A : contact with this subject. This is the reason why we present u3 v(E + p) u2 1 2 ¡ u u3 ¡ 2 (¡ ¡ 1) 2 a paper that is very detailed in the construction and imple- 1 u1 EXACT SOLUTION OF THE 1D RIEMANN PROBLEM IN NEWTONIAN AND RELATIVISTIC HYDRODYNAMICS 29 The initial data of the Riemann problem is defined as fol- coordinates (x; t) with the combination (x ¡ x0)=t; it can be lows seen that such behavior implies that the following conditions ½ hold [4] u ; x < x u = L 0 uR; x > x0; du1 du2 du3 i = i = i (5) r1 r2 r3 where uL and uR represent the values of the gas properties on a chamber at the left and at the right from an interface where i indicates the component of a given eigenvector. On the other hand, the Rankine Hugoniot conditions relate states between the two states at x = x0 that exists only at initial time. on both sides of a shock wave or a contact discontinuity The evolution of the initial data is described by the char- ¢F = V ¢u; (6) acteristic information of the system of equations, and this is why the properties of the Jacobian matrix are impor- which are simply jump conditions, where ¢u is the size of tant. The Jacobian matrix of the system of equations is the discontinuity in the variables, V is the velocity of either A(u) = @F=@u and explicitly reads the contact discontinuity or shock and ¢F is the change of 0 1 the flux across the discontinuity. 0 1 0 @ 1 2 A A = 2 (¡ ¡ 3)v (3 ¡ ¡)v ¡ ¡ 1 : 2.1. Contact discontinuity waves 3 ¡vE ¡E 3 2 (¡ ¡ 1)v ¡ ½ ½ ¡ 2 (¡ ¡ 1)v ¡v The contact discontinuity is described by the second eigen- Its eigenvalues satisfy the condition ¸1(u) < ¸2(u) < ¸3(u) vector and evolves with velocity ¸2. Let us then analyze the and are given by second eigenvector. In this case the Riemann invariant con- ditions read ¸ = v ¡ a (2) 1 d½ d(½v) dE = = : ¸ = v (3) 1 2 2 1 v 2 v ¸3 = v + a (4) These relations implies that d(½") = dv = 0, further imply- p ing that p = constant and v = constant across the contact where a = (@p=@½)js is the speed of sound in the gas, wave. In order to relate the two sides from the contact dis- which depends on the equation of state. For the ideal gas continuity we use the Rankine-Hugoniot conditions, which p = ½"(¡ ¡ 1), where ¡ is the ratio between the specific are given by heats at constantp pressure and volume ¡ = cp=cv, the speed of sound is a = (¡p=½). On the other hand, the eigenvec- ½LvL ¡ ½RvR = Vc(½L ¡ ½R); (7) tors of the Jacobian matrix read ½ v2 + p2 ¡ ½ v2 + p2 = V (½ v ¡ ½ v ); (8) 0 1 L L L R R R c L L R R 1 vL(EL + pL) ¡ vR(ER + pR) = Vc(vL(EL + pL) r1 = @ v ¡ a A ; H ¡ av ¡ vR(ER + pR)): (9) 0 1 0 1 1 1 Here Vc is the velocity of propagation of the contact discon- r2 = @ v A ; r3 = @ v + a A : tinuity. 1 2 0 2 v H + av The discontinuity travels at speed ¸ = v therefore the Vc = v. For this reason from Eq. (7) follows that vL = vR The eigenvectors r1; r2; r3 are classified in the following = Vc. As a consequence of this, Eq. (8) gives the condition way: pL = pR, which implies (9) is satisfied. Notice that no con- ² they are called genuinely non-linear when satisfy the dition on the density arises, which allows the density to be discontinuous. condition ru¸i ¢ ri(u) 6= 0. ² and linearly degenerate when ru¸i ¢ ri(u) = 0. 2.2. Rarefaction waves It happens that r2 is linearly degenerate and represents At this point we do not know the nature of waves 1 and 3, and a contact discontinuity, however the other two are genuinely we can assume they may be rarefaction waves. Once again non-linear. we use the Riemann invariant equalities, which for vectors 1 Depending on the particular region of the solution we and 3 read will use both the Riemann invariant conditions for rarefaction d½ d(½v) dE waves and the Rankine Hugoniot conditions for shocks and = = ; 1 v ¡ a H ¡ av contact discontinuities. The Riemann invariants are based on d½ d(½v) dE the self-similarity property of the solution in some regions, = = : in the sense that the solution depends on the spatial and time 1 v + a H + av Rev. Mex. Fis. E 59 (2013) 28–50 30 F. D. LORA-CLAVIJO, J. P. CRUZ-PEREZ,´ F. SIDDHARTHA GUZMAN,´ AND J. A. GONZALEZ´ Manipulation of these equalities results in the following a useful expression for vR arises equations "µ ¶ ¡¡1 # 2a p 2¡ d½ ½ v = v ¡ L R ¡ 1 : (19) = ¡ for ¸ ; (10) R L ¡ ¡ 1 p dv a 1 L d½ ½ The only unknown quantity is p .
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