1 Introduction
1.1 Current fields of research
1.2 Overview of the numerical methods
1.3 Plan of the review
2 Special Relativistic Hydrodynamics
2.1 Equations
$$\begin{array}{ccc}(\rho {u}^{\mu}{)}_{;\mu}& =& 0,\end{array}$$  (1) 
$$\begin{array}{ccc}{T}_{;\nu}^{\mu \nu}& =& 0,\end{array}$$  (2) 
$$\begin{array}{c}{T}^{\mu \nu}=\rho h{u}^{\mu}{u}^{\nu}+p{g}^{\mu \nu}.\end{array}$$  (3) 
$$\begin{array}{c}h=1+\varepsilon +\frac{p}{\rho},\end{array}$$  (4) 
$$\begin{array}{c}\frac{\partial \mathbf{u}}{\partial t}+\frac{\partial {\mathbf{F}}^{i}(\mathbf{u})}{\partial {x}^{i}}=0,\end{array}$$  (5) 
$$\begin{array}{c}\mathbf{u}=(D,{S}^{1},{S}^{2},{S}^{3},\tau {)}^{T}\end{array}$$  (6) 
$$\begin{array}{c}{\mathbf{F}}^{i}=(D{v}^{i},{S}^{1}{v}^{i}+p{\delta}^{1i},{S}^{2}{v}^{i}+p{\delta}^{2i},{S}^{3}{v}^{i}+p{\delta}^{3i},{S}^{i}D{v}^{i}{)}^{T}.\end{array}$$  (7) 
$$\begin{array}{ccc}D& =& \rho W,\end{array}$$  (8) 
$$\begin{array}{ccc}{S}^{i}& =& \rho h{W}^{2}{v}^{i},i=1,2,3,\end{array}$$  (9) 
$$\begin{array}{ccc}\tau & =& \rho h{W}^{2}pD,\end{array}$$  (10) 
$$\begin{array}{c}{v}^{i}=\frac{{u}^{i}}{{u}^{0}},\end{array}$$  (11) 
$$\begin{array}{c}W={u}^{0}=\frac{1}{\sqrt{1{v}^{i}{v}_{i}}}.\end{array}$$  (12) 
$$\begin{array}{c}p=p(\rho ,\varepsilon ).\end{array}$$  (13) 
2.2 SRHD as a hyperbolic system of conservation laws
2.3 Exact solution of the Riemann problem in SRHD
$$\begin{array}{c}I\to L{\mathcal{W}}_{\leftarrow}{L}_{*}\mathcal{C}{R}_{*}{\mathcal{W}}_{\to}R,\end{array}$$  (14) 
$$\begin{array}{c}{\mathcal{W}}_{\leftarrow (\to )}=\{\begin{array}{cc}{\mathcal{\mathcal{R}}}_{\leftarrow (\to )}& \text{for}{p}_{b}\le {p}_{a},\\ {\mathcal{S}}_{\leftarrow (\to )}& \text{for}{p}_{b}>{p}_{a},\end{array}\end{array}$$  (15) 
$$\begin{array}{c}\begin{array}{cccccc}\bullet I& \to & L{\mathcal{S}}_{\leftarrow}{L}_{*}\mathcal{C}{R}_{*}{\mathcal{S}}_{\to}R,& {p}_{L}<{p}_{L*}& =& {p}_{R*}>{p}_{R},\\ \bullet I& \to & L{\mathcal{S}}_{\leftarrow}{L}_{*}\mathcal{C}{R}_{*}{\mathcal{\mathcal{R}}}_{\to}R,& {p}_{L}<{p}_{L*}& =& {p}_{R*}\le {p}_{R},\\ \bullet I& \to & L{R}_{\leftarrow}{L}_{*}\mathcal{C}{R}_{*}{\mathcal{\mathcal{R}}}_{\to}R,& {p}_{L}\ge {p}_{L*}& =& {p}_{R*}\le {p}_{R}.\end{array}\end{array}$$  (16) 
$$\begin{array}{c}{v}_{R*}^{x}\left({p}_{*}\right)={v}_{L*}^{x}\left({p}_{*}\right)={v}_{*}^{x},\end{array}$$  (17) 
$$\begin{array}{c}{v}_{S*}^{x}\left(p\right)=\{\begin{array}{cc}{\mathcal{\mathcal{R}}}^{S}\left(p\right),& \text{if}p\le {p}_{S},\\ {\mathcal{S}}^{S}\left(p\right),& \text{if}p>{p}_{S},\end{array}\end{array}$$  (18) 
$$\begin{array}{c}\frac{d{v}^{x}}{dp}=\pm \frac{1}{\rho h{W}^{2}{c}_{s}}\frac{1}{\sqrt{1+g({\xi}_{\pm},{v}^{x},{v}^{t})}},\end{array}$$  (19) 
$$\begin{array}{c}g({\xi}_{\pm},{v}^{x},{v}^{t})=\frac{\left({v}^{t}{)}^{2}\right({\xi}_{\pm}^{2}1)}{(1{\xi}_{\pm}{v}^{x}{)}^{2}},\end{array}$$  (20) 
$$\begin{array}{c}{\xi}_{\pm}=\frac{{v}^{x}(1{c}_{s}^{2})\pm {c}_{s}\sqrt{(1{v}^{2})[1{v}^{2}{c}_{s}^{2}({v}^{x}{)}^{2}(1{c}_{s}^{2})]}}{1{v}^{2}{c}_{s}^{2}},\end{array}$$  (21) 
$$\begin{array}{c}{v}^{t}={h}_{S}{W}_{S}{v}_{S}^{t}{\left(\frac{1({v}^{x}{)}^{2}}{{h}^{2}+({h}_{S}{W}_{S}{v}_{S}^{t}{)}^{2}}\right)}^{1/2},\end{array}$$  (22) 
$$\begin{array}{c}{W}^{2}d{v}^{x}=\pm \frac{{c}_{s}}{\gamma p}dp=\pm \frac{{c}_{s}}{\rho}d\rho ,\end{array}$$  (23) 
$$\begin{array}{c}{\mathcal{\mathcal{R}}}^{S}\left(p\right)=\frac{(1+{v}_{S}){A}_{\pm}\left(p\right)(1{v}_{S})}{(1+{v}_{S}){A}_{\pm}\left(p\right)+(1{v}_{S})},\end{array}$$  (24) 
$$\begin{array}{c}{A}_{\pm}\left(p\right)={\left(\frac{\sqrt{\gamma 1}c\left(p\right)}{\sqrt{\gamma 1}+c\left(p\right)}\frac{\sqrt{\gamma 1}+{c}_{sS}}{\sqrt{\gamma 1}{c}_{sS}}\right)}^{\pm \frac{2}{\sqrt{\gamma 1}}},\end{array}$$  (25) 
$$\begin{array}{c}c\left(p\right)={\left(\frac{\gamma (\gamma 1)p}{(\gamma 1){\rho}_{S}(p/{p}_{S}{)}^{1/\gamma}+\gamma p}\right)}^{1/2}.\end{array}$$  (26) 
$$\begin{array}{c}{\mathcal{S}}^{S}\left(p\right)=\left({h}_{S}{W}_{S}{v}_{S}^{x}\pm \frac{p{p}_{S}}{j\left(p\right)\sqrt{1{V}_{\pm}(p{)}^{2}}}\right){\left[{h}_{S}{W}_{S}+(p{p}_{S})\left(\frac{1}{{\rho}_{S}{W}_{S}}\pm \frac{{v}_{S}^{x}}{j\left(p\right)\sqrt{1{V}_{\pm}(p{)}^{2}}}\right)\right]}^{1},\end{array}$$  (27) 
$$\begin{array}{c}{V}_{\pm}\left(p\right)=\frac{{\rho}_{S}^{2}{W}_{S}^{2}{v}_{S}^{x}\pm j(p{)}^{2}\sqrt{1+{\rho}_{S}^{2}{W}_{S}^{2}(1({v}_{S}^{x}{)}^{2})/j(p{)}^{2}}}{{\rho}_{S}^{2}{W}_{S}^{2}+j(p{)}^{2}}.\end{array}$$  (28) 
$$\begin{array}{c}j\left(p\right)=\sqrt{\frac{{p}_{S}p}{\frac{{h}_{S}^{2}h(p{)}^{2}}{{p}_{S}p}\frac{2{h}_{S}}{{\rho}_{S}}}},\end{array}$$  (29) 
$$\begin{array}{c}{h}^{2}{h}_{S}^{2}=\left(\frac{h}{\rho}+\frac{{h}_{S}}{{\rho}_{S}}\right)(p{p}_{S}).\end{array}$$  (30) 
$$\begin{array}{c}\left(1+\frac{(\gamma 1)({p}_{S}p)}{\gamma p}\right){h}^{2}\frac{(\gamma 1)({p}_{S}p)}{\gamma p}h+\frac{{h}_{S}({p}_{S}p)}{{\rho}_{S}}{h}_{S}^{2}=0.\end{array}$$  (31) 
$$\begin{array}{c}{v}^{t}={h}_{S}{W}_{S}{v}_{S}^{t}{\left(\frac{1({v}^{x}{)}^{2}}{{h}^{2}+({h}_{S}{W}_{S}{v}_{S}^{t}{)}^{2}}\right)}^{1/2}.\end{array}$$  (32) 

$$\begin{array}{c}\mathbf{u}=\mathbf{u}\left(\frac{x{x}_{0}}{t{t}_{0}};{\mathbf{u}}_{L},{\mathbf{u}}_{R}\right),\end{array}$$  (33) 
3 HighResolution ShockCapturing Methods
3.1 Relativistic PPM
$$\begin{array}{c}{\widehat{\mathbf{F}}}^{RPPM}=\mathbf{F}(\mathbf{u}(0;{\mathbf{u}}_{L},{\mathbf{u}}_{R}\left)\right),\end{array}$$  (34) 
3.2 Relativistic Glimm's method
$$\begin{array}{c}{\mathbf{u}}_{j+\frac{1}{2}}^{n+\frac{1}{2}}=\mathbf{u}\left(\frac{(j+{\xi}_{n})\Delta x}{(n+\frac{1}{2})\Delta t};{\mathbf{u}}_{j}^{n},{\mathbf{u}}_{j+1}^{n}\right),\end{array}$$  (35) 
3.3 Twoshock approximation for relativistic hydrodynamics

3.4 Roetype relativistic solvers
$$\begin{array}{c}{\widehat{\mathbf{F}}}^{Roe}=\frac{1}{2}\left(\mathbf{F}\left({\mathbf{u}}_{L}\right)+\mathbf{F}\left({\mathbf{u}}_{R}\right){\sum}_{p}\left{\stackrel{~}{\lambda}}^{\left(p\right)}\right{\stackrel{~}{\alpha}}^{\left(p\right)}{\stackrel{~}{\mathbf{r}}}^{\left(p\right)}\right),\end{array}$$  (36) 
$$\begin{array}{c}{\stackrel{~}{\alpha}}^{\left(p\right)}={\stackrel{~}{\mathbf{l}}}^{\left(p\right)}\cdot ({\mathbf{u}}_{R}{\mathbf{u}}_{L}),\end{array}$$  (37) 
$$\begin{array}{c}\stackrel{~}{\mathbf{w}}=\frac{{\mathbf{w}}_{L}+{\mathbf{w}}_{R}}{{k}_{L}+{k}_{R}},\end{array}$$  (38) 
$$\begin{array}{c}\mathbf{w}=\left(k{u}^{0},k{u}^{1},k{u}^{2},k{u}^{3},k\frac{p}{\rho h}\right),\end{array}$$  (39) 
$$\begin{array}{c}{k}^{2}=\sqrt{g}\rho h,\end{array}$$  (40) 
3.5 Falle and Komissarov upwind scheme
$$\begin{array}{c}\frac{\partial \mathbf{v}}{\partial t}+\mathcal{A}\frac{\partial \mathbf{v}}{\partial x}=0,\end{array}$$  (41) 
$$\begin{array}{c}{\mathbf{v}}_{L*}={\mathbf{v}}_{L}+{b}_{L}{\mathbf{r}}_{L}^{},{\mathbf{v}}_{R*}={\mathbf{v}}_{R}+{b}_{R}{\mathbf{r}}_{R}^{+},\end{array}$$  (42) 
$$\begin{array}{c}{\mathbf{v}}_{*}={\mathbf{v}}_{L}+{\sum}_{{\stackrel{~}{\lambda}}^{\left(p\right)}<0}{\stackrel{~}{\alpha}}^{\left(p\right)}{\stackrel{~}{\mathbf{r}}}^{\left(p\right)},\end{array}$$  (43) 
$$\begin{array}{c}{\mathbf{v}}_{*}={\mathbf{v}}_{R}{\sum}_{{\stackrel{~}{\lambda}}^{\left(p\right)}>0}{\stackrel{~}{\alpha}}^{\left(p\right)}{\stackrel{~}{\mathbf{r}}}^{\left(p\right)},\end{array}$$  (44) 
$$\begin{array}{c}{\stackrel{~}{\alpha}}^{\left(p\right)}={\stackrel{~}{\mathbf{l}}}^{\left(p\right)}\cdot ({\mathbf{v}}_{R}{\mathbf{v}}_{L}),\end{array}$$  (45) 
$$\begin{array}{c}{\widehat{\mathbf{F}}}^{FK}=\mathbf{F}(\mathbf{u}({\mathbf{v}}_{*}({\mathbf{v}}_{L},{\mathbf{v}}_{R})\left)\right).\end{array}$$  (46) 
3.6 Relativistic HLL method (RHLLE)
$$\begin{array}{c}{\mathbf{u}}^{HLL}\left(\frac{x}{t};{\mathbf{u}}_{L},{\mathbf{u}}_{R}\right)=\{\begin{array}{cc}{\mathbf{u}}_{L}& \text{for}x<{a}_{L}t,\\ {\mathbf{u}}_{*}& \text{for}{a}_{L}t\le x\le {a}_{R}t,\\ {\mathbf{u}}_{R}& \text{for}x>{a}_{R}t,\end{array}\end{array}$$  (47) 
$$\begin{array}{c}{\mathbf{u}}_{*}=\frac{{a}_{R}{\mathbf{u}}_{R}{a}_{L}{\mathbf{u}}_{L}\mathbf{F}\left({\mathbf{u}}_{R}\right)+\mathbf{F}\left({\mathbf{u}}_{L}\right)}{{a}_{R}{a}_{L}},\end{array}$$  (48) 
$$\begin{array}{c}{\widehat{\mathbf{F}}}^{HLL}=\frac{{a}_{R}^{+}\mathbf{F}\left({\mathbf{u}}_{L}\right){a}_{L}^{}\mathbf{F}\left({\mathbf{u}}_{R}\right)+{a}_{R}^{+}{a}_{L}^{}({\mathbf{u}}_{R}{\mathbf{u}}_{L})}{{a}_{R}^{+}{a}_{L}^{}},\end{array}$$  (49) 
$$\begin{array}{c}{a}_{L}^{}=min\{0,{a}_{L}\},{a}_{R}^{+}=max\{0,{a}_{R}\}.\end{array}$$  (50) 
$$\begin{array}{c}{a}_{R}=\frac{\overline{v}+{\overline{c}}_{s}}{1+\overline{v}{\overline{c}}_{s}},{a}_{L}=\frac{\overline{v}{\overline{c}}_{s}}{1\overline{v}{\overline{c}}_{s}},\end{array}$$  (51) 
3.7 Artificial wind method
3.8 Marquina's flux formula
3.9 Symmetric TVD, ENO schemes with nonlinear numerical dissipation
4 Other Developments
4.1 Van Putten's approach
$$\begin{array}{c}\mathbf{u}(t,x)={\mathbf{u}}_{0}\left(t\right)+{\mathbf{u}}_{1}(t,x),\mathbf{F}(t,x)={\mathbf{F}}_{0}\left(t\right)+{\mathbf{F}}_{1}(t,x).\end{array}$$  (52) 
$$\begin{array}{c}\frac{\partial {{\mathbf{u}}_{1}}^{*}}{\partial t}+{\mathbf{F}}_{1}=0,\end{array}$$  (53) 
$$\begin{array}{c}\frac{d{\mathbf{F}}_{0}}{dt}=0.\end{array}$$  (54) 
$$\begin{array}{c}{{\mathbf{u}}_{1}}^{*}(t,x)={\int}^{x}{\mathbf{u}}_{1}(t,y)dy.\end{array}$$  (55) 
4.2 Relativistic SPH
$$\begin{array}{c}\frac{d{N}_{a}}{dt}={\sum}_{b}{\nu}_{b}({\mathbf{v}}_{a}{\mathbf{v}}_{b})\cdot {\nabla}_{a}{W}_{ab},\end{array}$$  (56) 
$$\begin{array}{c}\frac{d{\widehat{\mathbf{S}}}_{a}}{dt}={\sum}_{b}{\nu}_{b}\left(\frac{{p}_{a}}{{N}_{a}^{2}}+\frac{{p}_{b}}{{N}_{b}^{2}}+{\mathbf{\Pi}}_{ab}\right)\cdot {\nabla}_{a}{W}_{ab},\end{array}$$  (57) 
$$\begin{array}{c}\frac{d{\widehat{\tau}}_{a}}{dt}={\sum}_{b}{\nu}_{b}\left(\frac{{p}_{a}{\mathbf{v}}_{a}}{{N}_{a}^{2}}+\frac{{p}_{b}{\mathbf{v}}_{b}}{{N}_{b}^{2}}+{\mathbf{\Omega}}_{ab}\right)\cdot {\nabla}_{a}{W}_{ab},\end{array}$$  (58) 
$$\begin{array}{c}N=\frac{D}{{m}_{0}}\end{array}$$  (59) 
$$\begin{array}{c}\widehat{\mathbf{S}}\equiv \frac{\mathbf{S}}{N}={m}_{0}hW\mathbf{v}\end{array}$$  (60) 
$$\begin{array}{c}\widehat{\tau}\equiv \frac{\tau}{N}+{m}_{0}={m}_{0}hW\frac{p}{N}\end{array}$$  (61) 
$$\begin{array}{c}{\mathbf{\Pi}}_{ab}=\frac{K{v}_{sig}({\widehat{\mathbf{S}}}_{a}^{*}{\widehat{\mathbf{S}}}_{b}^{*})\cdot \mathbf{j}}{{\overline{N}}_{ab}},\end{array}$$  (62) 
$$\begin{array}{c}\mathbf{j}=\frac{{\mathbf{r}}_{ab}}{\left{\mathbf{r}}_{ab}\right}\end{array}$$  (63) 
$$\begin{array}{c}{\widehat{\mathbf{S}}}^{*}={m}_{0}h{W}^{*}\mathbf{v},\end{array}$$  (64) 
$$\begin{array}{c}{W}^{*}=\frac{1}{\sqrt{1(\mathbf{v}\cdot \mathbf{j}{)}^{2}}}.\end{array}$$  (65) 
$$\begin{array}{c}{\mathbf{\Omega}}_{ab}=\frac{K{v}_{sig}({\widehat{\tau}}_{a}^{*}{\widehat{\tau}}_{b}^{*})\mathbf{j}}{{\overline{N}}_{ab}},\end{array}$$  (66) 
4.3 Relativistic beam scheme
5 Summary of Methods


Code  Basic characteristics 
Artificial viscosity  
AVmono [48, 121, 187]  Nonconservative formulation of the RHD equations 
(transport differencing, internal energy equation);  
artificial viscosity extra term in the momentum flux;  
monotonic secondorder transport differencing;  
explicit time stepping.  
cAVimplicit [213]  Nonconservative formulation of the RHD equations; 
internal energy equation;  
consistent formulation of artificial viscosity;  
adaptive mesh and implicit time stepping.  
cAVmono [8]  Nonconservative formulation of the RHD equations 
(transport differencing, internal energy equation);  
consistent bulk scalar and tensorial artificial viscosity;  
monotonic secondorder transport differencing;  
explicit time stepping.  
Flux corrected transport  
FCTlw [75]  Nonconservative formulation of the RHD equations 
(transport differencing, equation for $\rho hW$ );  
explicit secondorder Lax–Wendroff scheme with FCT algorithm.  
SHASTAc  FCT algorithm based on SHASTA [31] ; 
[256, 67, 68, 244, 246]  advection of conserved variables. 
van Putten's approach  
van Putten [287]  Ideal RMHD equations in constraintfree, divergence form; 
evolution of integrated variational parts of conserved quantities;  
smoothing algorithm in numerical differentiation step;  
leapfrog method for time stepping.  
Smooth particle hydrodynamics  
SPHAV0  Specific internal energy equation; 
[172] (SPH0), [148]  artificial viscosity extra terms in momentum and energy equations; 
secondorder time stepping  
([172] : predictorcorrector; [148] : RK method).  
SPHAV1 [172] (SPH1)  Time derivatives in SPH equations include variations in smoothing 
length and mass per particle;  
Lorentz factor terms treated more consistently;  
otherwise same as SPHAV0.  
SPHAVc [172] (SPH2)  Total energy equation; 
otherwise same as SPHAV1.  
SPHcAVc [261]  RHD equations in conservation form; 
consistent formulation of artificial viscosity.  
SPHRSc [51]  RHD equations in conservation form; 
dissipation terms constructed in analogy to terms in Riemann  
solverbased methods.  
Code  Basic characteristics 
SPHRSgr [204]  GRSPH conservation equations [202] ; 
dissipation terms as in [51] .  
6 Test Bench
6.1 Relativistic shock heating in planar, cylindrical and spherical geometry
$$\begin{array}{c}{\varepsilon}_{2}={W}_{1}1.\end{array}$$  (67) 
$$\begin{array}{c}\sigma =\frac{\gamma +1}{\gamma 1}+\frac{\gamma}{\gamma 1}{\varepsilon}_{2},\end{array}$$  (68) 
$$\begin{array}{c}{V}_{s}=\frac{(\gamma 1){W}_{1}\left{v}_{1}\right}{{W}_{1}+1}.\end{array}$$  (69) 
$$\begin{array}{c}\rho (t,r)={\left(1+\frac{\left{v}_{1}\rightt}{r}\right)}^{\alpha}{\rho}_{0},\end{array}$$  (70) 

6.2 Propagation of relativistic blast waves
Problem 1  Problem 2  
Left  Right  Left  Right  
$p$  $13.33$  $0.00$  $1000.00$  $0.01$  
$\rho $  $10.00$  $1.00$  $1.00$  $1.00$  
$v$  $0.00$  $0.00$  $0.00$  $0.00$  
${v}_{shell}$  $0.72$  $0.960$  
${w}_{shell}$  $0.11t$  $0.026t$  
${v}_{shock}$  $0.83$  $0.986$  
${\sigma}_{shock}$  $5.07$  $10.75$  
6.2.1 Problem 1


6.2.2 Problem 2

6.2.3 Collision of two relativistic blast waves

7 Applications
7.1 Astrophysical jets
7.2 Gammaray bursts (GRBs)