Numerical Methods for Two-Fluid Dispersive Fast MHD Phenomena

Transcription

1 Numeical Methods fo Two-Fluid Dispesive Fast MHD Phenomena B. Sinivasan 1,, A. Hakim 2, and U. Shumlak 3 1 Applied Mathematics and Plasma Physics, Los Alamos National Laboatoy, Los Alamos, NM Tech-X Copoation, 5621 Aapahoe Avenue Suite A, Boulde, CO Aeospace and Enegetics Reseach Pogam, Univesity of Washington, Seattle, WA Abstact. The finite volume wave popagation method and the finite element Runge- Kutta discontinuous Galekin (RKDG) method ae studied fo applications to balance laws descibing plasma fluids. The plasma fluid equations exploed ae dispesive and not dissipative. The physical dispesion intoduced though the souce tems leads to the wide vaiety of plasma waves. The dispesive natue of the plasma fluid equations exploed sepaates the wok in this pape fom pevious publications. The lineaized Eule equations with dispesive souce tems ae used as a model equation system to compae the wave popagation and RKDG methods. The numeical methods ae then studied fo applications of the full two-fluid plasma equations. The two-fluid equations descibe the self-consistent evolution of electon and ion fluids in the pesence of electomagnetic fields. It is found that the wave popagation method, when un at a CFL numbe of 1, is moe accuate fo equation systems that do not have dispaate chaacteistic speeds. Howeve, if the oscillation fequency is lage compaed to the fequency of infomation popagation, souce splitting in the wave popagation method may cause phase eos. The Runge-Kutta discontinuous Galekin method povides moe accuate esults fo poblems nea steady-state as well as poblems with dispaate chaacteistic speeds when using highe spatial odes. 1 Intoduction Thee ae a numbe of equation systems that ae eithe hypebolic o contain hypebolic pats. Homogeneous, hypebolic equation systems ae witten as consevation laws of the fom [1, 2] Coesponding autho. addess: bsinivasan@lanl.gov Q + F =0, (1.1) t Global Science Pepint

2 2 whee Q R m epesents the m conseved vaiables and F R m d epesents fluxes in d spatial diections. Fo all unit vectos ω R d the flux Jacobian, (F ω)/ Q, has eal eigenvalues and a complete set of ight eigenvectos. Some homogeneous, hypebolic equation systems include the Eule equations and magnetohydodynamic (MHD) equations. Inhomogeneous, hypebolic equation systems ae descibed by balance laws of the fom Q + F = S, (1.2) t whee S R m epesents the souce tems. The souce Jacobian fo equation Eq. (1.2) is S Q. The pesence of eal eigenvalues in the souce Jacobian esults in an equation system that contains diffusive souces. The Navie-Stokes equations and the 10-moment fluid equations [3] ae examples of inhomogeneous, hypebolic equation systems containing diffusive souce tems. Fo inhomogeneous, hypebolic equation systems descibed by Eq. (1.2), the pesence of imaginay eigenvalues in the souce Jacobian esults in an equation system that contains dispesive souces. The two-fluid plasma model is a system of inhomogeneous, hypebolic equations containing dispesive souce tems. The dispesive souce tems aise fom the physical popeties of the plasma medium. Dispesive souce tems pesent a unique challenge fo numeical algoithms because low-ode, explicit-time-stepping schemes can be unstable when applied to the wave equation leading to numeical dispesion [4]. The physical dispesion can be difficult fo numeical schemes to captue and can be difficult to distinguish fom the numeical dispesion o noise. In this pape, numeical methods fo solving inhomogeneous, hypebolic equations containing dispesive souce tems ae investigated fo accuacy and computational effot. Hypebolic consevation laws can have discontinuous solutions even if the initial conditions ae smooth, and this makes the appoximation of the solution difficult. Fist ode upwind methods ae needed to effectively captue such discontinuities. Howeve, fist ode methods ae highly diffusive in smooth egions. Second ode extensions can be constucted which both esolve the discontinuities and povide bette accuacy in smooth egions. Smooth nonlinea solutions can achieve second ode accuacy when using Godunov s method with second ode coections [5] even though the method is fomally fist ode accuate, e.g. in Sec of Ref. [2]. Ref. [5] povides poof of second ode accuacy fo smooth poblems including the case with souce tems (Sec. 7 of Ref. [5]). Fo a cetain class of finite volume methods such as the high-esolution wave popagation method, the solution can be diffusive when eigenvalues of the flux Jacobian have a significant spead. Highe than second ode extensions ae thus needed. In this pape a highe-ode (>O(1)) finite volume method [2, 5] and a spectally accuate finite element method [6] ae applied to the plasma fluid equations. The wave popagation method belongs to the class of finite volume methods. The domain is discetized into cells and the cell aveages of the conseved vaiables ae evolved.

3 3 A Riemann poblem is solved at each cell edge and is used to compute the numeical fluxes. Using these computed fluxes, the solution is then updated in the cells connected to that edge. To achieve highe than fist ode accuacy a linea econstuction of the waves is pefomed at the edge which allows up to a second ode accuate solution. Limites ae applied to educe the spatial ode in the egions of lage gadients wheeas egions of small gadients maintain highe-ode. Fo simulations with stong shocks, limites often educe the solution to fist ode aound the shocks. The Runge-Kutta discontinuous Galekin (RKDG) method is a finite element method. While the wave popagation method is second ode, the discontinuous Galekin method achieves highe spatial ode accuacy by expanding the solution in basis functions defined locally in each cell. The ode of the polynomial detemines the spatial ode of the method. As in the case of the wave popagation method, Riemann poblems ae solved at the cell edge. The data fo the Riemann poblem ae computed using the basis function expansion. A Runge-Kutta time stepping method is used to advance the solution in time. An oveview of the development of discontinuous Galekin (DG) methods is povided by Cockbun, Kaniadakis and Shu in Ref. [7]. DG methods wee oiginally developed in the famewok of neuton tanspot equations by Reed and Hill [8] fo solving linea hypebolic equations. The DG method was then applied to non-linea advectiondominated hypebolic systems by Cockbun and Shu [9] who used the high-ode TVD Runge-Kutta time integation that was developed by Shu [10] with DG and developed local pojection slope-limites to balance the spuious oscillations in egions of shap gadients. This was extended to highe spatial odes and multiple dimensions fo applications to non-linea systems such as the Eule equations. Next came the evolution of the DG method fo applications to advection-diffusion equation systems [6, 11, 12], such as the Navie-Stokes equations. The DG method has also been applied to Maxwell s equations [13,14] whee the electomagnetic oscillations need to be appopiately esolved and to single-fluid MHD equations [15] whee the plasma waves need to be appopiately esolved. The application of DG methods to nonlinea dispesive equations, specifically the Koteweg-de Vies equation, is exploed in Ref. [16] whee DG methods ae shown to be advantageous in the pesence of apid oscillations and ae capable of simultaneously captuing oscillations and discontinuous fonts in the solution. Zhang and Shu have compaed the discontinuous Galekin method to spectal finite volume methods in Ref. [17] fo linea one-dimensional hypebolic equations and state that the spectal finite volume method has lage eos than the DG method on the same mesh. The plasma fluid equations studied in this pape diffe fom peviously published wok because the dispesive natue of the non-linea balance laws oiginates fom the puely dispesive souce tems. The appopiate teatment of advection and dispesion tems and the complications that aise fom the pesence of significant dispesion in plasma models povide a unique set of compaisons between the wave popagation and the RKDG methods. Thee is vey little liteatue egading the numeical solution of hypebolic equation systems that contain dispesive souce tems without the pesence of any explicit dissipation [4, 18]. Hakim, Loveich and Shumlak [4] implement the

4 4 high-esolution wave popagation method fo the two-fluid plasma model studied in this pape. Loveich and Shumlak [18] implement the Runge-Kutta discontinuous Galekin method fo the two-fluid plasma model. Neithe of these efeences addess the effect of the physical dispesive souce tems of the two-fluid plasma model on the accuacy of the numeical methods. The two-fluid equations descibe the self-consistent evolution of electon and ion fluids in the pesence of electomagnetic fields. The two-fluid plasma equation system is diffeent fom Eule equations o Navie-Stokes equations that have been peviously studied with the discontinuous Galekin method. The two-fluid plasma equations ae dispesive and not dissipative. The dispesion is not a numeical atifact. The dispesive natue is a physical effect that leads to the wide vaiety of plasma waves. The plasma waves in this pape ae fast MHD waves that esult fom esolving the ion and electon plasma fequencies in the pesence of electomagnetic waves. Mathematically, the dispesive effects ae geneated fom the souce tems of the two-fluid plasma equations descibed in Section 2.2. The dispesive natue of these equation systems sets them apat fom pevious publications. The study of the appopiateness of the wave popagation and the discontinuous Galekin methods to accuately captue these physical dispesive effects is a main point of the manuscipt. Section 2 descibes the full two-fluid plasma model showing the basic equations and pefoming a souce tem analysis to highlight the physically expected dispesion bought about by the souces. Section 3 biefly descibes the high-esolution wave popagation method and Section 4 biefly descibes the RKDG method. Section 5 pesents a convegence study using a linea advection equation to quantify the spatial ode fo the RKDG and wave popagation methods. Section 6 exploes a benchmak poblem that uses the wave popagation method and the RKDG method. The benchmak poblem uses the Eule equations with the addition of dispesive souce tems. This povides a model equation system fo the two-fluid plasma model. Following a study of the wave popagation and RKDG methods fo the benchmak poblem, the numeical methods ae applied to the two-fluid plasma poblems in Sec. 7. The simulations ae pefomed using the WARPX (Washington Appoximate Riemann Plasma) code developed at the Univesity of Washington. WARPX is witten in C++ and povides a geneal famewok fo developing paallel computational physics algoithms. The wave popagation and the RKDG methods ae implemented fo abitay hypebolic consevation laws and use the same undelying appoximate Riemann solves and flux calculation methods wheeve appopiate. This allows a fai compaison of the methods since they both use the same optimized famewok to solve the equations. All simulations pesented in this pape ae pefomed on ectangula, egula meshes.

5 5 2 Full Two-Fluid Plasma Model The full two-fluid plasma model [19] used in this section is deived by taking moments of the Boltzmann equation and teating the electons and ions as two sepaate fluids. The esulting Eule equations ae used to evolve the electon and ion fluids while Maxwell s equations ae used to evolve the electomagnetic tems. The equations that esult have homogeneous, hypebolic pats and inhomogeneous souce tems. 2.1 Govening Equations fo the Full, Two-Fluid Plasma Model The equations descibed hee ae the five-moment equations that esult fom taking the zeoth, fist and second moments of the Boltzmann equation. These moment equations ae closed with an equation of state. Assumptions used ae isotopic pessue, no heat flux and no fictional foces. The electons and ions ae each descibed by the Eule equations with souce tems coupling the fluids and the fields. In balance law fom, the fluid equations ae ρ s u s t ρ s t + (ρ su s ) =0 (2.1) + (ρ s u s u s + p s I) = ρ sq s (E+u s B) m s (2.2) ϵ s t + ((ϵ s+p s )u s ) = ρ sq s u s E m s (2.3) whee subscipt, s, denotes electon o ion species. q s is the species chage, m s is the species mass, ρ is the mass density, u is the velocity, E is the electic field, B is the magnetic field, p is the pessue and ϵ is the total enegy. The enegy is defined as ϵ s p s γ ρ su 2 s. (2.4) Maxwell s equations ae used to evolve the electic and magnetic fields. B + E =0 t (2.5) 1 E c 2 t B = µ 0J (2.6) E = ϱ c ε 0 (2.7) B =0 (2.8) whee ϱ c and J ae the chage density and the cuent density defined by ϱ c s J s q s m s ρ s (2.9) q s m s ρ s u s. (2.10)

6 6 The souce tems of Eq. (2.2) contain the Loentz foces on the electons and ions. These souce tems couple the fluid equations to the electomagnetic fields. The Loentz foces act as body foces on the electons and ions. The evolving electomagnetic souce tems can make the equation set and the solutions athe complicated. The full two-fluid equations ae applied to the one- and two-dimensional Z-pinch equilibium. Applications and esults of the two-fluid plasma model ae pesented in Sec. 7.3, 7.4 and Souce Tems of the Two-Fluid Plasma Model The two-fluid plasma model has physical dispesion that comes about fom the pesence of dispesive souce tems. The souce Jacobian fo equation Eq. (1.2) is J S = S Q. The fist thee eigenvalues of the souce Jacobian ae 0,±iω p whee ω 2 p = ω 2 pe+ω 2 pi. The plasma fequency is defined as ω ps = n s q 2 s ε 0 m s, (2.11) whee subscipt s epesents each species (electons and ions). The emaining six eigenvalues ae oots of the polynomial with espect to λ, +B 2 2 i 1 ( M 2 λ λ 2 +ω 2 pe+ωpi) [ 2 B 4 λ 2 i 4+ M2 λ 2( λ 2 +ω 2 pe+ω 2 pi ( (M 2 λ 4 +λ 4 +2M 2 ω 2 peλ 2 + M 2 ω 4 pe+ω 4 pi +2 λ 2 Mω 2 pe ) ω 2 pi ) 2 )] =0 (2.12) whee M is the electon-to-ion mass atio, i is the ion chage-to-mass atio, and B is the magnitude of the magnetic field. All non-zeo eigenvalues, λ, ae imaginay. Since the souce Jacobian has only imaginay eigenvalues, the waves of the two-fluid model ae not damped. This plasma model is not diffusive but instead is dispesive with undamped oscillations simila to the model equation system using Eule equations with dispesive souce tems exploed in Sec. 6. This dispesive natue of the two-fluid plasma model sets the wok in this pape apat fom pevious publications. The wave popagation and RKDG methods ae studied fo thei abilities to captue these physical dispesions accuately. Explicit methods ae often unstable when such oscillations ae pesent because efining the gid can excite waves of smalle wavelengths making it difficult to captue the dispesions accuately. The egime of applicability of the wave popagation and RKDG methods is exploed in the pesence of such dispesions. 3 High Resolution Wave Popagation Method The high esolution wave popagation method can be applied to balance laws of the fom Eq. (1.2). This method is descibed in detail fo two-fluid plasma equations in Ref. [4].

7 7 The wave popagation method belongs to the class of Godunov methods which ely on the solution of Riemann poblems. The essential idea is as follows. The domain is discetized into cells and the solution in each cell is assumed to be epesented by aveages. At each cell inteface the solution is econstucted and will, in geneal, be discontinuous. This discontinuity is used as an initial condition fo a Riemann poblem. The solution of the Riemann poblem gives the conseved vaiables at the inteface which ae then used to compute numeical fluxes. Once the fluxes ae known the solution in each cell is updated by tallying how much flux flows into the cell. In one dimension, up to second ode accuacy can be achieved by pefoming a linea econstuction of the waves needed to compute the numeical fluxes at the cell inteface. In multiple dimensions high esolution tansvese coections ae included which account fo flow that is tansvese to the coodinate axes. Afte solving the Riemann poblem at each cell inteface to detemine the positive- and negative-going fluctuations, a second tansvese Riemann poblem is solved to compute the tansvese fluctuations as detailed in Ref. [2]. With these tansvese coections, the method is fomally second ode accuate and is stable to CFL numbe of unity even when using Godunov splitting, as descibed in Section 17.5 of Ref. [2]. The wave popagation update fomula in 1-dimension is given by Q n+1 i = Qi n t [ A + Q x i 1/2 +A ] Q i+1/2 t ( ) [ F] x i+1/2 [ F] i 1/2, (3.1) whee A + Q i 1/2 and A Q i+1/2 ae the positive- and negative-going fluctuations, and [ F], is the coection flux descibed in Ref. [2]. The souce tems fo the wave popagation method ae handled using souce splitting. A Godunov splitting is used hee as detailed in Ref. [2] whee the homogeneous hypebolic equation is solved fist to update the conseved quantities, followed by a Runge-Kutta update to advance the solution with the souce tems. The souce tems of the two-fluid system ae paticulaly challenging as they epesent undamped oscillations, i.e., the Jacobian of the souce tems has puely imaginay eigenvalues. Such souces add physical dispesion to the system which can be difficult to esolve. Fo a discussion elevant to the two-fluid system see [4]. 4 Discontinuous Galekin Method The RKDG method is a finite element method. The RKDG method achieves highe spatial ode by expanding the solution in polynomial basis functions. The balance law descibed in Eq. (1.2) is multiplied by a set of basis functions and is integated ove the element. The conseved vaiable is defined as a linea combination of the basis functions. In this pape tenso poducts of the Legende polynomials ae chosen as basis functions.

8 8 This allows the constuction of methods of abitay spatial ode. Riemann poblems ae solved at each inteface to compute the inteface fluxes needed in the algoithm. In geneal, simple appoximate solves can be used with the RKDG method when using high spatial odes since local fluctuations ae accuately epesented by the high-ode basisfunctions [6]. In contast, the wave popagation method needs a moe accuate Riemann solve to avoid diffusive eos. The RKDG method update fomula in 1-dimension is given by dq dt = F i+1/2v (x i+1/2 ) F i 1/2 v (x i 1/2 ) + 1 dv (x) Fdx+ 1 v (x)sdx. (4.1) x x I i dx x I i whee Q epesents the expansion coefficients and v epesents the polynomial basis functions descibed in Ref. [6]. A 3 d ode total vaiation diminishing (TVD) Runge-Kutta method is used fo the time integation as discussed in Ref. [6]. This makes the RKDG method an explicit finite element method. When the tempoal ode is geate than the spatial ode,, the time step is esticted fo numeical stability using CFL 1/(2 1). When the tempoal ode is smalle than, the CFL numbe depends on the accuacy of the Runge- Kutta method [6] and can be moe estictive. Stong stability peseving Runge-Kutta (SSPRK) methods have been exploed in ecent yeas and optimal high-ode SSP methods have been shown to povide highe tempoal accuacy fo linea and nonlinea poblems with the ability to use lage time steps [20, 21]. While highe-ode SSP time integation schemes could be easily extended fo use with the DG method used in this pape, thee is no benefit gained by using the highe tempoal accuacy fo the plasma poblems exploed hee because of the elatively smooth tempoal evolution. This is veified by exploing the two-fluid model with the 2-dimensional axisymmetic Z-pinch in Sec. 7.5 using high-ode SSPRK methods to show no qualitative diffeences fom the 3 d ode TVD RK scheme o even the 2 nd ode TVB RK scheme. A 3 d ode TVD Runge-Kutta method is well suited fo time integation fo the poblems exploed in this pape in tems of both accuacy and computational effot. An effective esolution is intoduced to povide a fai compaison metic among the numeical methods. The effective esolution takes into account the numbe of degees of feedom, i.e. the numbe of equations being solved fo a given spatial ode. The wave popagation method solves m equations in N g cells, whee m efes to the numbe of conseved vaiables in a given equation system. The degees of feedom fo the wave popagation method is theefoe N g fo each vaiable. The RKDG method solves m d equations in N g cells, whee d is the numbe of dimensions. The RKDG method has N g degees of feedom fo each vaiable, m, in each dimension, d. Theefoe, fo the RKDG method, the effective esolution is defined as N g. The RKDG method can poduce lage oscillations in the solution when shap gadients ae pesent. As in the case of the wave popagation method, limites ae applied. Fo the wave popagation method the limites ae applied to the waves, howeve, fo the RKDG method limites ae applied diectly to the conseved vaiables o the chaacteis-

9 9 tic vaiables. A modified minmod limite is used hee whee the linea tems ae checked fo oscillations and the high ode tems ae set to zeo if the linea tems need to be limited [22]. Developing high-ode limites is a challenging eseach poblem fo the DG method. Some pospective high-ode limites ae exploed in Refs. [23]& [24]. 5 Convegence Study The linea advection of a one-dimensional Gaussian pulse q(x,0) = e 10(x 1.5)2 is used to numeically detemine the spatial ode fo the wave popagation and RKDG methods. This benchmak poblem does not contain any souce tems and is included to demonstate the ode of accuacy and convegence fo the wave popagation and RKDG methods. The linea advection equation is q t + q =0. (5.1) x Peiodic bounday conditions on a domain 1 < x < 5 ae used. Afte popagating the pulse one peiod though the domain, the l 2 nom of the eo is computed by compaing to the exact solution. The spatial ode of the method is computed by measuing the dependence of the l 2 -nom on the gid spacing. Figue 1 shows the measued l 2 -noms of the solutions obtained fo diffeent gid esolutions with a fixed time step that is stable fo all the numeical methods tested. ( t= 003.) The slopes measued fom the linea egions of the plot ae listed in the second column of Table 1. The 8 th ode RKDG solution conveges to the analytical solution apidly so the linea behavio is only obseved at lowe esolutions. The table shows that the computed ode of convegence exceeds the fomal ode of the methods fo this linea poblem with a fixed t. The fixed time step isolates the effect of the spatial ode. Figue 2 shows the l 2 -noms of the solutions obtained fo diffeent gid esolutions with a vaiable time step that is chosen based on the maximum allowable CFL numbe fo each method at each esolution. The maximum CFL numbe fo each method is shown in the 4 th column on Table 1 based on the spatial and tempoal odes [6]. The slopes measued fom the linea egions of the plot ae listed in the thid column of Table 1. Fo vaiable t, the computed ode of convegence fo the highe-ode RKDG methods is limited to appoximately 3 due to the thid-ode time-integation method used with the RKDG method. 6 Eule Equations with Dispesive Souce Tems Due to the complexity of the full two-fluid plasma system it is difficult to investigate the effects of dispesion on the full non-linea physics. In this section a simple model is intoduced that allows fo dispesion to be included with the Eule equations in the fom of dispesive souce tems. This povides a model equation system fo the two-fluid

10 10 Method Ode Ode CFL (fixed t) (maximum t) (maximum t) WAVE RKDG 2 nd ode RKDG 3 d ode RKDG 4 th ode RKDG 8 th ode Table 1: Slopes of l 2 -nom vs x to detemine the ode of accuacy of the methods fo the linea advection equation. Column 2 shows that the obseved spatial ode fo this poblem exceeds the fomal ode of the methods when using a fixed time-step fo all spatial odes. Howeve, column 3 shows the actual ode obtained when using the vaiable time-step with the maximum allowable CFL numbe (based on the spatial and tempoal accuacy of the scheme). The obseved tempoal ode fo the highe ode RKDG methods is limited by the thid-ode, time-integation method used fo all RKDG solutions no m l x wave kdg2 kdg3 kdg4 kdg8 Figue 1: Log-log plot of l 2 -noms of solution as a function of x fo the linea advection poblem using a fixed time step to isolate the effect of the spatial ode fo all the numeical methods - wave popagation, 2 nd, 3 d, 4 th, and 8 th ode RKDG. The slopes of the lines ae tabulated in Table 1.

11 no m l wave kdg2 kdg3 kdg4 kdg x Figue 2: Log-log plot of l 2 -noms of solution as a function of x fo the linea advection poblem using the maximum allowable CFL numbe to set the time step fo each of the numeical methods - the wave popagation, 2 nd, 3 d, 4 th, and 8 th ode RKDG. The slopes of the lines ae tabulated in Table 1. plasma equations because it geneates physical dispesions but has simple souce tems and a closed-fom analytical solution exists fo the linea case. This equation system models the quasineutal ion cycloton waves, which ae dispesive waves, in a unifom plasma with a magnetic field that is constant in space and time. The momentum equation includes the foce fom a unifom tansvese magnetic field, which poduces dispesive effects. The momentum equation, Eq. (2.2), simplifies to ρ ( u t +u x u ) + p = nqu B (6.1) x = ρω c u ˆb. (6.2) whee ρ = m i n is the ion mass density, u = (u,v) is the fluid velocity, p is the pessue, B is a unifom magnetic field, ˆb is the unit vecto of B and ω c = qb/m i is the cycloton fequency. The Eule equations of gas dynamics with dispesive souce tems ae witten in one dimensional non-consevative fom as ρ u ρ 0 0 ρ u t v + 0 u 0 1/ρ u 0 0 u 0 x v = p 0 γp 0 u p 0 vω c uω c 0 (6.3) whee γ is the adiabatic index. The eigenvalues of the souce Jacobian ae 0,0,±iω c. These compae well with the puely imaginay eigenvalues of the souce Jacobian of the

12 12 two-fluid plasma model descibed in Section 2.2, the fist thee of which ae 0,±iω p. The undamped oscillations epesented by the pesence of the cycloton fequency in the dispesive Eule equations povide a suitable benchmak model fo the undamped oscillations of the two-fluid plasma model. The non-zeo imaginay eigenvalues indicates that the system has undamped, non-popagating oscillations which combined with the sound wave leads to dispesive waves ω n = ± ( k 2 nc 2 s +ω 2 c ) 1/2 (6.4) whee c s γp 0 /ρ 0 is the speed of sound and k n is the wave numbe. To study the ability of the methods to captue the dispesion coectly, a poblem in the linea egime is solved. The equations ae lineaized about a static unifom equilibium with density ρ 0 and pessue p 0. Assuming a petubed solution of the fom f = f 0 + f 1, whee f {ρ,u,v,p}, and whee f 1 (x,t) is of the fom f 1 (x,t) = n=1 ˆf n e i(k nx+ω n t) (6.5) lineaized equations ae obtained iω n ik n ρ ρ 1 0 iω n ω c ik n /ρ 0 u 1 0 ω c iω n 0 v 1 = 0. (6.6) 0 ik n γp 0 0 iω n p 1 whee ρ 1, u 1, v 1, and p 1 depend on n. This system of linea algebaic equations has nontivial solutions only if ω n and k n satisfy the dispesion elation given by Eq. (6.4). The dispesion elation is nonlinea in k n leading to the dispesion of waves as they popagate though the fluid. To initialize the simulation the fluid is petubed with a velocity u 1 (x) = u 0 1 N i n=0 2n+1 eik nx (6.7) with k n = 2π(2n+1) and ω n computed fom Eq. (6.4) and u 0 1 is a constant. As N, Eq. (6.7) epesents a step function fo the inteval [0,1]. With the petubation of Eq. (6.7) the exact solution fo the lineaized velocity u(x,t) is given by ũ(x,t) = N n=0 u 0 1 2n+1 sin(k nx+ω n t). (6.8) The test poblem is initialized using the exact solution fo all petubed vaiables with the velocity given by Eq. (6.8). Figue 3 shows the initial condition fo u 0 1 = 10 8, N = 9

13 13 x1e u x Figue 3: Initial condition with N = 9 that is used to appoximate the step function. This initial condition is used fo all the esults obtained in this section. Method l 2 -nom Computational time CFL to t =1 WAVE RKDG RKDG RKDG RKDG Table 2: l 2 -nom of velocity to quantify accuacy fo each method, and computational time equied to advance the solution to t =1 to quantify computational effot fo the dispesive Eule system using ω c =10. using γ=2, ω c =10 and ρ 0 = p 0 =1. Fo these values the sound speed is given by c s = 2. Peiodic bounday conditions ae applied on a domain 0 < x < 1. A CFL numbe of 1 is used fo the wave popagation method and 1/(2 1) is used fo the RKDG method. The tempoal ode of the RKDG method is 3 d ode fo this poblem. Limites ae not applied in eithe method and the solutions at t =3 ae compaed to the exact solution. Figues 4 and 5 compae the analytical solution to the wave popagation method and to the RKDG method. The numbe of gid elements, N g, is adjusted with the spatial ode,, of the RKDG method so the effective esolution, N g, emains constant. Accuacy is measued by taking an l 2 -nom. These figues along with Table 2 show that the wave popagation method is moe accuate than the 2 nd, 3 d and 5 th ode RKDG methods while all methods have the same effective esolution. The 8 th ode RKDG solution with only 12 cells, howeve, is moe accuate than the wave popagation method. The computational time equied to advance the solution fom t = 0 to t = 1 fo each method is pesented in Table 2. Each method has diffeent CFL stability limit as shown in the 4 th column of Table 2 based on Table 2.2 in Ref. [6] whee the spatial and tempoal odes ae taken into account to detemine the maximum CFL value. The solution of the

14 14 u u x1e x1e-8 x Analytical wave_100 kdg2_50 kdg3_ x Figue 4: Velocity at t=3 fo an effective esolution of 100 cells fo the wave popagation and RKDG methods, i.e., 100 cells fo wave popagation, 50 fo 2 nd ode RKDG and 33 fo 3 d ode RKDG. c s = 2 and ω c = 10. The bottom plot has an expanded scale to show the details of the solution.

15 15 u u x1e x1e-8 x Analytical wave_100 kdg5_20 kdg8_ x Figue 5: Velocity at t = 3 with 100 cells fo wave popagation, 20 fo 5 th ode RKDG and 12 fo 8 th ode RKDG. c s = 2 and ω c =10. The bottom plot has an expanded scale to show the details of the solution.

16 16 u x1e x1e-8 x Analytical wave_100 kdg3_33 u x Figue 6: Velocity at t=3 with 100 cells fo wave popagation and 33 fo 3 d ode RKDG. These ae fo c s = 2 and ω c = 50. Using a lage ω c leads to phase eos fo the wave popagation method while the lowe ode RKDG method fo the same effective esolution is diffusive but has no phase eo. The bottom plot has an expanded scale to show the details of the solution. wave popagation method is moe accuate as compaed to the RKDG solutions while using less computational effot fo low ω c. Howeve, when ω c is inceased, the wave popagation method exhibits phase eos in the solution. Figue 6 shows that the 3 d ode RKDG solution using 33 cells is moe diffusive than the wave popagation solution at 100 cells, but the RKDG solutions do not have phase eos even at lowe odes. Figue 7 displays esults fo the wave popagation method with 100 cells, the 8 th ode RKDG with 12 cells and the 16 th ode RKDG with 6 cells. Inceasing the gid esolution of the wave popagation method fom 100 cells to 500 cells educes the phase eo, as shown in Fig. 8, and going to even highe esolution eliminates it. The computational time equied to advance the solution fom t = 0 to t = 1 fo each method fo ω c =50 is pesented in Table 3. Table 3 shows that the 16 th ode RKDG solution with only 6 cells is moe accuate than the 500 cell wave popagation method. The 16 th ode RKDG method with only 6 cells uses the same computational effot as the wave

17 17 u x1e x1e-8 x Analytical wave_100 kdg8_12 kdg16_6 u x Figue 7: Velocity at t = 3 with 100 cells fo wave popagation, 12 fo 8 th ode RKDG and 6 fo 16 th ode RKDG. These ae fo c s = 2 and ω c = 50. Using a lage ω c leads to phase eos fo the wave popagation method. The bottom plot has an expanded scale to show the details of the solution. Method l 2 -nom Computational time CFL to t =1 WAVE WAVE RKDG RKDG RKDG Table 3: l 2 -nom of velocity to quantify accuacy fo each method, and computational time equied to advance the solution to t =1 to quantify computational effot fo the dispesive Eule system using ω c =50.

18 18 u x1e x1e-8 x Analytical wave_100 wave_500 u x Figue 8: Velocity at t = 3 with 100 cells as compaed to 500 cells fo the wave popagation method. These ae fo c s = 2 and ω c = 50. The bottom plot has an expanded scale to highlight the small phase eo that is pesent even with 500 cells when a highe cycloton fequency is used. Method l 2 -nom Computational time CFL to t =1 WAVE WAVE RKDG Table 4: l 2 -nom of velocity to quantify accuacy fo each method, and computational time equied to advance the solution to t =1 to quantify computational effot fo the dispesive Eule system using ω c =100.

19 19 x1e u x1e-8 x Analytical wave_100 wave_500 dg10_10 u x Figue 9: Velocity at t = 3 with 100 cells and 500 cells fo the wave popagation method as compaed to 10 cells fo the 10 th ode RKDG method. These ae fo c s = 2 and ω c =100. The bottom plot has an expanded scale to highlight the phase eo that is pesent even with 500 cells when a highe cycloton fequency is used.

20 20 popagation method with 500 cells. Fo lage ω c, the RKDG method povides a moe accuate solution even when it is un at a lowe effective esolution using high spatial ode. Fo compaable accuacy with lage ω c, the RKDG method uses less computational effot as compaed to the wave popagation method. The phase eos in the wave popagation method ae caused by the lage souce tems compaed to the advection tems that esult fom inceasing ω c. This hypothesis is suppoted by measuing lage phase eos when the souce tem is inceased by setting ω c = 100 as compaed to the ω c = 50 solution in Fig. 8. The ω c =100 esults ae shown in Fig. 9 and Table 4. In paticula, the eo fo the 500 cell solution is lage fo the ω c = 100 solution than fo the ω c = 50 solution. When the magnitude of the souce tem becomes lage compaed to the advection tems in the equation system, the wave popagation method poduces phase eos. Inceasing the souce tem stength fo a given esolution inceases the eo. Table 4 shows that a 10 th ode RKDG method with 10 cells has highe accuacy and uses less computational effot than a 500 cell wave popagation method when ω c = 100. Hence, the pope handling of souce tems becomes citical. The wave popagation method uses the souce tem splitting descibed in Sec. 3, and this splitting leads to the phase eos. The chaacteistic oscillation peiod caused by the souce tems is τ c = 2π/ω c fo the solution. The chaacteistic time fo infomation to popagate is τ s = x/c s. Fo the oscillation to be well esolved, τ c must be sufficiently lage than τ s. Fo the case of the wave popagation method with 100 cells, this equiement is violated because τ c = while τ s = 71, which is not sufficient to esolve τ s. The lack of pope sampling leads to the phase eo seen in Figs. 7 and 9. The chaacteistic fequency of the souces intoduces ωc 1 time-scales that must be esolved in addition to the othe time-scales in the system. The explicit time-step must be sufficiently small fo pope sampling of the souce fequency such that t < ωc 1. To futhe suppot that the souce splitting causes the phase eos, an unsplit implicit souce tem update is implemented fo the wave popagation method using 100 cells with ω c = 50. The implicit souce tem update is descibed by ( Q(t+ t) = I t ) 1 [ 2 J S Q(t) tl(q(t))+ t ] 2 J SQ(t) whee L epesents the flux update. Figue 10 shows that while the implicit souce tem solution is moe diffusive, the phase eos ae eliminated. Exploing unsplit souce tem handling fo the wave popagation method fo equation systems with puely dispesive souce tems, without the diffusive natue of an implicit souce tem update, could make the wave popagation method moe obust to such phase eo poblems. A von Neumann analysis is pefomed to specifically quantify the stability condition fo the souce tem update fo the wave popagation method. It is noted that as long as the condition (6.9) t 2 2 ω c (6.10)

21 21 u x1e Analytical - wave_ wave_100_implicit Figue 10: Velocity at t=3 with 100 cells fo the wave popagation method using souce splitting vesus using an unsplit implicit souce tem update. These ae fo c s = 2 and ω c =50. The implicit souce update is included to pove that souce splitting is esponsible fo the phase eos. Howeve, the implicit solution is subject to sevee diffusion. is satisfied, the wave popagation method with a Runge-Kutta souce advance is stable in the pesence of lage ω c. If a t is chosen such that the stability condition in Eq. (6.10) is satisfied, and the time-step accounts fo the additional ωc 1 time-scale, the wave popagation solution with 100 cells becomes diffusive. The wave popagation method geneally becomes diffusive with CFL numbes less than 1. This pesents a numeical difficulty in esolving the physical dispesions accuately while minimizing diffusive eos and equies a highe gid esolution fo the wave popagation method. Using highe ode spatial epesentations with the RKDG method solves this poblem with less computational effot and geate accuacy in the pesence of lage souce tems. 7 Two-Fluid Plasma Simulations The two-fluid plasma model descibed in Sec. 2 is investigated using the wave popagation and RKDG methods. The two-fluid plasma model contains 18 equations when using the eo coection potentials fo the puely hypebolic Maxwell s equations [25]. The eo coection potentials povide 2 additional conseved vaiables to the full two-fluid equation system. They ae included as a modification to Maxwell s equations and allow divegence eos to be advected out of the domain at a specified chaacteistic speed. The challenge with this model lies in esolving all the waves popagating though the domain with speeds anging fom the speed of sound to the speed of light. The eo coection speed fo the puely hypebolic Maxwell s equations can be even lage than the speed of light. This section contains fou applications of the two-fluid plasma model - a 1-dimensional soliton popagation, an axisymmetic two-fluid pulse in 1-dimension, an axisymmetic Z-pinch equilibium in 1-dimension and a petubed axisymmetic Z- pinch equilibium in 2-dimensions.

22 Bounday Conditions fo Axisymmetic Poblems Ghost cells ae used to specify the bounday conditions fo both the wave popagation and the RKDG methods. Using ghost cells allows fo the application of Diichlet bounday conditions whee the vaiables cay a fixed edge value and Neumann bounday condition whee the gadient at the bounday is specified. Fo axisymmetic poblems, the axis bounday condition is implemented at = 0. Appopiate bounday conditions at the axis ae found by assuming the vaiables ae analytical about the axis and pefoming a seies expansion about = 0. Radial and azimuthal vecto components ae set to zeo at the axis. Scala vaiables and axial vecto components have no gadient nomal to the axis. To implement the axis bounday condition using ghost cells, the scala vaiables and axial vecto components ae copied into the ghost cells while the adial and azimuthal vecto components ae copied ove to the ghost cells with a negative sign. The poblem of singulaities at the axis does not aise hee because a modal implementation of the DG method is used instead of a nodal implementation. Fo poblems with conducting wall bounday conditions, the nomal velocity, the nomal magnetic field, and the tangential electic fields go to 0 at a conducting wall. To implement this bounday condition using ghost cells, all vaiables ae copied fom the adjacent domain cells to the ghost cells while evesing the signs of the nomal velocity, nomal magnetic field and tangential electic fields. The emaining vaiables ae extapolated fom the domain. The coefficients of the RKDG method must be teated appopiately at the boundaies. As mentioned in Sec. 4, Legende polynomials ae chosen as the basis functions. The physical bounday conditions ae implemented with consideation to the polynomial basis functions of the RKDG method. Fo example, to implement a Diichlet bounday value of zeo, the coefficients of all even basis functions in the ghost cells ae set to the negative of the adjacent domain cells, and the coefficients of all odd basis functions in the ghost cells ae set to the same values as the adjacent domain cells. To make the aveage value at the bounday zeo, the sign is flipped in the ghost cell fo all the coefficients of the Legende polynomial that have even exponents of x since these polynomials ae the even basis functions. The polynomials with odd exponents of x ae the odd basis functions and do not equie a change in sign in the ghost cells. The implementation of the zeo gadient at the bounday uses the same pocedue with opposite signs fo the coefficients of the even and odd basis functions. 7.2 Two-Fluid Plasma Soliton in 1-Dimension The two-fluid plasma model, Eq. ( ), is applied to one-dimensional soliton popagation [26] whee a pulse is initialized in the ion and electon densities and pessues as shown in Fig. 11. The fluid pessues ae intialized using fluid tempeatues such that T i = T e = 1 and B z = 1. All othe fluid and field vaiables ae initialized to zeo.

23 ρ i x Figue 11: Initial ion mass density fo the two-fluid plasma soliton. Ion pessue, electon mass density, and electon pessue have the same pofile. Method l 2 -nom Computational time CFL to t =40 WAVE WAVE RKDG RKDG RKDG RKDG Table 5: l 2 -nom of ion mass density to quantify accuacy fo each method and computational time equied to advance the solution to t = 40 to quantify computational effot fo the two-fluid soliton. The ion-to-electon mass atio is 25. The atio of the speed of light to the electon sound speed, c/c se =2, and the atio of the speed of light to the ion sound speed, c/c si =10. The speed of light is chosen such that it is the fastest speed in the system. Peiodic bounday conditions ae used. The wave popagation solution at a esolution of 5000 cells is chosen as the conveged solution and is used to compae the wave popagation method to the RKDG method. This conveged solution is compaed to the 3 d ode RKDG using 1000 cells to veify that both methods convege to the same solution. Simulations ae un to t = 40 whee time is nomalized by the speed of light tansit time acoss the domain. One full peiod of the ion soliton occus at t =100. Figue 12 and Table 5 show that fo the same effective esolution of 512 cells, i.e. wave popagation with 512 cells, 2 nd ode RKDG with 256 cells and 4 th ode RKDG with 128 cells, the RKDG method povides a moe accuate solution than the wave popagation method. Figue 13 shows that even when the esolution of the wave popagation method is doubled to 1024 cells, the solution is less accuate than the appoximate effective esolution of 512 cells using 3 d, 4 th, and 5 th ode RKDG methods.

24 24 ρ i Conveged wave_512 kdg2_256 kdg4_ x ρ i x Figue 12: The ion mass density, ρ i, is compaed fo the two-fluid plasma soliton using wave popagation and RKDG methods. The solution is shown at t = 40 with c = 1. The wave popagation method uses 512 cells, RKDG 2 nd ode uses 256 cells, and RKDG 4 th ode uses 128 cells so all methods have the same effective esolution. The bottom plot has an expanded scale to show the details of the solution. The wave popagation method has the lagest phase eos.

25 25 ρ i Conveged wave_1024 kdg3_171 kdg5_ x ρ i x Figue 13: The ion mass density, ρ i, is compaed fo the two-fluid plasma soliton using wave popagation and RKDG methods. The solution is shown at t=40 with c=1. The RKDG 3 d ode solution with 171 cells and the RKDG 5 th ode solution with 100 cells povide a moe accuate solution than the 1024 cell wave popagation method at double the effective esolution. The bottom plot has an expanded scale to show the details of the solution.

26 26 Phase eos in the wave popagation method solution ae evident in the expanded scale plots of Figs. 12 and 13. These phase eos occu in the waves that ae popagating away fom the initial pulse and esult fom souce splitting just as with the quasineutal ion cycloton waves exploed in Sec. 6. The computational time equied to advance the solution fom t = 0 to t = 40 fo each method is pesented in Table 5. Each method has diffeent CFL stability limit as shown in the 4 th column of Table 5 based on Table 2.2 in Ref. [6] whee the spatial and tempoal odes ae taken into account to detemine the maximum CFL value. When using 1024 cells, the wave popagation method takes 2 times the computational effot as compaed to the 4 th ode RKDG method using 128 cells, and 4 times the computational effot as compaed to the 5 th ode RKDG method using 100 cells. Thee does not seem to be an obvious tend in the CPU times fo the RKDG method. This nonmonotonic vaiability in the CPU times is attibuted to the stability condition fo the RKDG methods descibed in Ref. [6]. CFL 1/(2 1) is valid fo spatial ode,, as long as the tempoal ode is +1. Fo all DG spatial odes pesented hee, a 3 d ode Runge-Kutta time integation scheme is used because 2 nd ode Runge-Kutta time integation is unstable fo DG when > 2. In ode to use 3 d ode Runge-Kutta time integation with > 2, the maximum allowable CFL numbe is moe estictive as descibed in Ref. [6]. Fo this poblem, the RKDG method povides a moe efficient solution when computational effot is taken into account as shown in Table 5. While inceasing the gid esolution eliminates the phase eos in the wave popagation method, the RKDG method is moe computationally efficient fo this poblem. Two-dimensional applications of a two-fluid soliton poduce simila esults. 7.3 Axisymmetic Two-Fluid Plasma Pulse in 1-Dimension Following a soliton popagation application, the two-fluid plasma model, Eq. ( ), is applied to an axisymmetic one-dimensional poblem whee a pulse is initialized in the axial magnetic field, B z, as shown in Fig. 14. The electon and ion densities and pessues ae initially constant thoughout the domain with all othe fluid and electomagnetic tems initialized to 0. The ion-to-electon mass atio is 2.5 to minimize negative pessue eos and allow fo faste evolution. The atio of the speed of light to the electon sound speed, c/c se = 35, and the atio of the speed of light to the ion sound speed, c/c si = 55. Axis bounday conditions ae used on the left edge of the domain while conducting wall bounday conditions ae used on the ight edge. The bounday conditions ae teated in the manne descibed in Sec The wave popagation solution at a esolution of 10,000 cells is chosen as the conveged solution and is used to compae the wave popagation method to the RKDG method. This conveged solution is compaed to the 3 d ode RKDG using 1000 cells and it is veified that both methods convege to the same solution. The fluid azimuthal velocity, v ϕ, shows the lagest vaiation among the methods. Fo this eason, the electon azimuthal velocity is chosen fo the compaisons. Simulations ae un to t=0.8 whee time

27 Bz Figue 14: Initial condition fo axial magnetic field fo the axisymmetic two-fluid plasma pulse. v φ Conveged wave_100 kdg3_33 kdg5_ v φ Figue 15: The electon fluid azimuthal velocity, v ϕ, is compaed fo the axisymmetic two-fluid plasma pulse using wave popagation and RKDG methods. The solution is shown at t=0.8 with c=1. The wave popagation method uses 100 cells, RKDG 3 d ode uses 33 cells and RKDG 5 th ode uses 20 cells so all methods have appoximately the same effective esolution. The bottom plot has an expanded scale to show the details of the solution. At the same effective esolution, the wave popagation method pefoms pooest.

28 28 v φ Conveged wave_200 kdg5_ v φ Figue 16: The electon fluid azimuthal velocity, v ϕ, is compaed fo the axisymmetic two-fluid plasma pulse using wave popagation and RKDG methods. The solution is shown at t=0.8 with c=1. The RKDG 5 th ode solution uses 10 cells with an effective esolution that is 1/4 that of the wave popagation method which uses 200 cells. The bottom plot has an expanded scale to show the details of the solution. Even with a lowe effective gid esolution that is 1/4 the esolution of the wave popagation method, the RKDG method povides a moe accuate solution than the wave popagation method. Method l 2 -nom Computational time CFL to t =0.4 WAVE WAVE RKDG RKDG RKDG Table 6: l 2 -nom of electon azimuthal velocity to quantify accuacy fo each method, and computational time equied to advance the solution to t = 0.4 to quantify computational effot fo the two-fluid pulse.