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recently , the gravitational waves which are the solutions of the einstein equations and are the part of the most interesting astrophysical events were observed by ligo @xcite . the einstein equations are sets of the nonlinear partial differential equations with constraints , numerical calculations are the realistic tools to study these equations for the actual physical events in the universe such as the gravitational waves . analyzing the einstein equations numerically is called `` numerical relativity '' , these studies made a substantial contribution to observe the gravitational wave events . these discoveries must open new physical fields to survey of lots of astrophysical events . therefore , it is important to improve numerical techniques to make high accuracy numerical results . although , there are some researches of the suitable numerical schemes of the einstein equations(e.g . @xcite ) , it is not enough to perform precisely simulations . however , it would be hardly to make best discretized equations of the einstein equations because of the strong nonlinearity . in addition , some of the characteristics in the continuous system are often lost in the process of transforming the continuous equation to the discretized one . there are some discretized schemes of preserving properties of the continuous equations . one of the schemes is the discrete variational derivative method ( dvdm ) . dvdm was proposed and has been extended by furihata , mori , matsuo , and yaguchi @xcite , and this method is one of the structure - preserving methods @xcite . the method of formulating a discrete system using dvdm is similar to the variational principle . in the variational principle , we provide a generator function such as the lagrange density and we can obtain a continuous system using the functional derivative of the function . on the other hand , in dvdm , we provide a discretized generator function of the system and we can obtain a discrete system using the discrete variational derivative . the crank - nicolson scheme ( cns ) and the runge - kutta scheme are general methods of discretizing equations , these schemes often use in numerical relativity . on the other hand , dvdm is a specific method of preserving constraints and diffusion characteristics in the continuous system . since the einstein equations include the constraints , dvdm can provide a discretized einstein equations with constraint preserving properties .    before we try to make the discretized einstein equations using dvdm , we studied the discretized maxwell s equations using the method because of the complexities of the einstein equations@xcite . the maxwell s equations are classified the hyperbolic equations with constraints , the characters of the system are similar to those of the einstein equations . in @xcite , the authors studied the numerical stabilities for the maxwell s equations using dvdm , and showed that dvdm is effective to perform stable simulations in the system .    in this letter , indices such as @xmath0 run from @xmath1 to @xmath2 . we use the einstein convention of summation of repeated up - down indices . now , we review the processes of the standard method of discretized equations using general schemes and the new way using dvdm . the discrete values of the variable @xmath3 are defined as @xmath4 . the upper index @xmath5 and lower index @xmath6 denote the time components , and space components , respectively . the details of this are in ref . we adopt only the canonical formulation in this letter , so that these processes are only shown in the case that the generator function is the hamiltonian density . first , we introduce the process of the general case of constructing the discrete equations such as cns and the runge - kutta scheme . ( a ) : we set a hamiltonian density @xmath7 with the variables @xmath8 and the canonical conjugate momenta @xmath9 . ( b ) : the dynamical equations are given as @xmath10 then we discretize the continuous equations following the ideas of the schemes such as cns and the runge - kutta scheme . next , the processes of discrete equations using dvdm are as follows : ( a@xmath11 ) : we set a discrete hamiltonian density @xmath12 with the discrete variables @xmath13 and the discrete canonical conjugate momenta @xmath14 . ( b@xmath11 ) : the calculation of the discretized system using dvdm is @xmath15 where @xmath16 and @xmath17 are calculated as @xmath18 the lagrangian density of the einstein equations is given by @xmath20 where @xmath21 is lapse function , @xmath22 is the determinant of the three - metric @xmath23 , and @xmath24 is the three - scalar curvature . @xmath25 is the extrinsic curvature , @xmath26 . the word gr means the general relativity . with the variables @xmath23 and the canonical momenta @xmath27 , the hamiltonian formulation of the einstein equations can be constructed . this formulation was given by arnowitt , deser , and misner ( adm ) , this is the first successful decomposition of the einstein equations to space and time @xcite . if we discretize the equations of the formulation with dvdm , we face to two difficulties . first is discretizing of @xmath22 . the variational of @xmath22 in continuous case is @xmath28 it is hard to make the above relation in discretized case . then , we use the conformal transformation such as @xmath29 to vanish @xmath22 in the lagrangian density . these relations are derived from following two conditions : ( i ) the lagrangian density @xmath30 does not include @xmath31 , ( ii ) the evolution equation of @xmath32 does not include @xmath31 . if these conditions are satisfied , the @xmath31 is not also included in the expression of the hamiltonian density @xmath33 derived from the legendre transformation of the lagrangian density . thus , the canonical formulation using @xmath33 does not include @xmath31 . with , the lagrangian density can be expressed as @xmath34 where @xmath35 is the scalar curvature of @xmath32 , @xmath36 is the connection of @xmath32 , + @xmath37 , and @xmath38 . note that the expressions of @xmath30 and @xmath39 are different , but they are same , that is to say @xmath40 . the relation can be derived in straightforward . then the hamiltonian density is defined as @xmath41 next is construction of suitable discretization of the @xmath24 . this is defined as in continuous case such as @xmath42 note that there are the same relations between @xmath35 , @xmath36 , and @xmath32 . this means that the conformal transformation does not overcome the difficulty of @xmath24 . therefore we treat only @xmath19 to simplify @xmath24 in this letter . if @xmath33 is the function of only @xmath19 , then the density can be expressed as @xmath43 the evolution equations are @xmath44 and the constraint equation is @xmath45 in the adm formulation , the constraint values are the hamiltonian constraint @xmath46 and the momentum constraints @xmath47 , thus there are four constraints . however , there is only one equation of constraint in the above set of equations . this is because we treat the case of @xmath33 as only @xmath19 function . if we treat the case of @xmath48 as @xmath49 function , there are four constraint equations in the new formulation . the evolution equation of can be derived as @xmath50 then the hamiltonian constraint is satisfied in the evolutions . we call the evolution equation of constraint as constraint propagation ( cp ) . we set the discrete hamiltonian density as @xmath51 then , the discretized evolution equations with dvdm are @xmath52 the discretized cp is derived as @xmath53 thus , the discretized cp of dvdm is the same as the continuous case .      to compare with the numerical stability of the discrete system using dvdm , we select the crank - nicolson scheme ( cns ) which is used in the numerical relativity . the discrete evolution equations with cns are @xmath54 and the discretized cp with the scheme is @xmath55 this is not equal zero in general because that the terms @xmath56 , @xmath57 , @xmath58 , and @xmath59 are not equal zero . therefore , the constraint values would be changed of initial values using the cns . we perform some simulations to study differences of the constraint violations using the two schemes . then , we adopt the initial condition as @xmath60 this is the kasner solution which is one of the exact solutions of the einstein equations . this solution expresses the expanding universe , and there is a singularity at @xmath61 . then the initial conditions of @xmath21 , @xmath23 , and @xmath25 are given by @xmath62 and using the relations of becomes as @xmath63 in addition , it is hardly to make constraint violations in the above condition . then , we add perturbations such as @xmath64 the other conditions in these simulations given by    * initial time : @xmath65 . * time steps : @xmath66 . * @xmath21(gauge condition ) : exact solution of . * boundary condition : periodic . there are @xmath67 time step variables in the discretized evolution equations using dvdm and cns ( - and - , respectively ) . in the simulations , the variables are calculated in iterative because of the nonlinearlity , then we set the numbers of times of iteration as fourth order in both schemes . and horizontal axis means time . the solid line is drawn by dvdm and the dotted line is by cns . the line of cns is grow up after initial time . on the other hand , the one of dvdm is horizontal in the simulations . [ fig : hamiltonain ] ]    fig.[fig : hamiltonain ] is the constraint violations using dvdm and cns . in this figure , the solid line which is drawn by dvdm is horizontal , and the dotted line which is drawn by cns is rising . this means that the values of the constraint @xmath48 of dvdm is unchanged in the evolution . on the other hand , that of cns is increasing from the initial values . therefore , these results are consistent with the results of analyzing cp in sec.[sec : discretizedeqs ] . we derived suitable discretized equations of the einstein equations using the discrete variational derivative method ( dvdm ) . in the process of the discretization , there are two difficulties . one is the existence of the determinant of @xmath23 , @xmath22 , in the lagrangian density , and the other is the complexity of the scalar curvature @xmath68 included in the density . for the existence of @xmath22 , we eliminated it by the conformal transformation of the variables . for the complexity of @xmath68 , we restricted the coordinate as only @xmath19 . then , we derived the discretized evolution equations of the variables and constraint in dvdm and the crank - nicolson scheme ( cns ) , showed the discrete constraint of dvdm is unchanged and the one of cns is changed . in addition , we performed some simulations using the discretized equations of the dvdm and cns . then , the constraint values using the dvdm were conserved . on the other hand , the values using cns were not conserved . these results are consistent with the analysis of the discretized evolution equations of the constraint using the dvdm and cns . therefore the dvdm is the validity to perform stable simulations in the numerical relativity .    in this letter , we restricted the coordinates to simplify the complexities of the scalar curvature @xmath24 . to perform realistic simulations , we should handle the @xmath24 in the coordinates @xmath49 . so we would like to try it in the future work . g. y. was partially supported by a waseda university grant for special research projects ( number 2016b-121 , 2016k-161 ) . 22 abbott , b. p. _ et al . _ ( ligo scientific collaboration ) , observation of gravitational waves from a binary black hole merger , phys . rev . lett . * 116 * 061102 ( 2016 ) . abbott , b. p. _ et al . _ ( ligo scientific collaboration and virgo collaboration ) , gw151226 : observation of gravitational waves from a 22-solar - mass binary black hole coalescence , phys . lett . * 116 * 241103 ( 2016 ) . r. richter and c. lubich , free and constrained symplectic integrators for numerical general relativity , class . quantum grav . * 25 * 225018 , ( 2008 ) . d. furihata and m. mori , a stable finite difference scheme for the cahn - hilliard equation based on the lyapunov functional , zamm z. angew . mech . * 76 * , 405 - 406 ( 1996 ) . d. furihata , finite difference schemes for @xmath69 that inherit energy conservation or dissipation property , j. comput . phys . * 156 * , 181 - 205 ( 1999 ) . t. matsuo and d. furihata , dissipative or conservative finite difference schemes for complex - values nonlinear partial differential equations , j. comput.phys . * 171 * , 425 - 447 ( 2001 ) . d. furihata and t. matsuo , discrete variational derivative method , crc press / taylor & francis , london , ( 2010 ) . y. aimoto , t. matsuo , and y. miyatake , a local discontinuous galerkin method based on variational structure , discrete contin . * 8 * , 817 - 832 ( 2015 ) . a. ishikawa and t. yaguchi , geometric investigation of the discrete gradient method for the webster equation with a weighted inner product , jsiam letters * 7 * , 17 - 20 ( 2015 ) . h. hairer and c. lubich , geometric numerical integration , springer - verlag , berlin heidelberg , ( 2010 ) . t. tsuchiya and g. yoneda , constraint - preserving scheme for maxwell s equations , arxiv : gr - qc/1610.04370 . r. arnowitt , s. deser , and c. w. misner , in gravitation : an introduction to current research , edited by l. witten , wiley , new york , ( 1962 ) ; r. arnowitt , s. deser , and c. w. misner , republication of : the dynamics of general relativity , gen . . gravit . * 40 * , 1997 - 2027 ( 2008 ) . r. m. wald , general relativity , the university of the chicago press , chicago , ( 1984 ) .
we propose a new numerical scheme of evolution for the einstein equations using the discrete variational derivative method ( dvdm ) . 
 we derive the discrete evolution equation of the constraint using this scheme and show the constraint preserves in the discrete level . 
 in addition , to confirm the numerical stability using this scheme , we perform some numerical simulations by discretized equations with the crank - nicolson scheme and with the new scheme , and we find that the new discretized equations have better stability than that of the crank - nicolson scheme .