High-Order Energy and Linear Momentum Conserving Methods for the Klein-Gordon Equation

Transcription

1 matematics Article Hig-Order Energy and Linear Momentum Conserving Metods for te Klein-Gordon Equation He Yang Department of Matematics, Augusta University, Augusta, GA 39, USA; Tel.: Received: 7 September 8; Accepted: 3 October 8; Publised: October 8 Abstract: Te Klein-Gordon equation is a model for free particle wave function in relativistic quantum mecanics. Many numerical metods ave been proposed to solve te Klein-Gordon equation. However, efficient ig-order numerical metods tat preserve energy and linear momentum of te equation ave not been considered. n tis paper, we propose ig-order numerical metods to solve te Klein-Gordon equation, present te energy and linear momentum conservation properties of our numerical scemes, and sow te optimal error estimates and superconvergence property. We also verify te performance of our numerical scemes by some numerical examples. Keywords: ig-order numerical metods; te Klein-Gordon equation; energy-conserving metod; linear momentum conservation; local discontinuous Galerkin metods; optimal error estimates; superconvergence. ntroduction Te Klein-Gordon equation is a widely used model in relativistic quantum mecanics. t is used to describe te free particle wave function. Many numerical metods, including finite difference metods [ 4], finite element metods [5] and spectral metods [6,7], ave been proposed to solve te Klein-Gordon equation. n particular, te autors in [] proposed a simple second order centered difference in time and space, an explicit sceme to solve te Klein-Gordon equation. Te finite difference scemes proposed in [,4] used te same treatment for second order spatial and temporal derivatives, but different treatment for te source term of te equation. n [3], a finite difference metod along wit an operator splitting metod was proposed to solve te equation on an unbounded domain. All of tese finite difference scemes ave second-order convergence in space and time. Some types of spectral metods were proposed in order obtain a iger order of convergence in space. n [6], a pseudo-spectral metod was proposed for te spatial discretization, and Crank-icolson or leap-frog metod was used for te temporal discretization. n [7], Legendre wavelets incorporated wit a spectral metod were used to solve Klein-Gordon and Sine-Gordon equations. As far as te family of finite element metods is concerned, Ref. [5] proposed Galerkin finite element metods for te Klein-Gordon equation wit omogeneous Diriclet boundary conditions. However, none of te aforementioned work focused on te conservation properties of energy and momentum. Te advantage of ig-order energy and linear momentum conserving metods is tat tey can lead to accurate and pysically relevant solutions wit relatively coarse mes. Moreover, te structure-preserving property is a significant factor wen udging te performance of numerical scemes [8]. n tis paper, we propose ig-order local discontinuous Galerkin LDG metods tat ave provable properties of energy and linear momentum conservation, optimal convergence and superconvergence. Discontinuous Galerkin DG metods belong to te family of finite element metods, and ave drawn great attention since te work by Cockburn and Su in [9 ]. LDG metods, as a special type of DG metod, were ten proposed to solve diffusion equations [3], partial differential Matematics 8, 6, ; doi:.339/mat6

2 Matematics 8, 6, of 7 equations wit ig spatial derivatives [4,5], and wave equations [6 9]. n tis work, we propose new metods for te Klein-Gordon equation based on te local discontinuous Galerkin metods because LDG metods ave some advantage of classical finite element metods. Tat is, it is easy to design LDG metods wit arbitrary order of convergence, and it is suitable for mes adaptivity and complex geometry. More importantly, DG metods can be designed to conserve mass and energy for certain systems of partial differential equations [,]. Te outline of tis paper is as follows. n Section, we present our semi-discrete LDG metods, prove te conservation properties of energy and linear momentum of te Klein-Gordon equation, and sow te optimal convergence of our numerical solutions. All of te teoretical results from tis section are based on our semi-discrete LDG scemes. n Section 3, we give rigorous proof about te superconvergence property of our numerical solution. n Section 4, we present two types of temporal discretizations and discuss teir properties. n Section 5, we conclude tis paper by performing some numerical experiments to test te optimal convergence, superconvergence and conservation properties. We can sow tat our numerical results are consistent wit our teoretical results.. Local Discontinuous Galerkin Discretization.. Semi-Discrete Local Discontinuous Galerkin Metods for te Klein-Gordon Equation Te original form of te Klein-Gordon equation is given by c t + m c φ, were φ is te wave function, c is te speed of ligt, m is te mass and is te Planck constant. Applying te following transformation x mc x, t mc t, ux, t φ x, t to Equation, we ave te non-dimensional Klein-Gordon equation u tt u + u. 3 n tis paper, we develop ig-order numerical metods to solve Equation 3 using te metod of lines. We first define te semi-discrete local discontinuous Galerkin sceme for a one-dimensional Klein-Gordon equation, and te multi-dimensional equation can be defined in te same manner. We consider te Klein-Gordon equation defined on a one-dimensional spatial domain [a, b] wit initial conditions ux, f x, u t x, gx and a periodic boundary condition. Before we present our numerical metods, we introduce te important notations tat we will need later. We use H m to denote te L -Sobolev space of order m wose equipped norm is represented by m. We use L instead of H, wen m, and its corresponding norm is denoted by. Similarly, we use W m, to represent te L -Sobolev space of order m wose equipped norm is denoted by m,. Wen m, we simply use to denote te L norm. Suppose we partition te spatial domain into subintervals, eac of wic is denoted by for,,...,. Let a x < x 3 < < x + b be te endpoints of tese subintervals, and [x, x + ] for all. For eac, let x x + x + be te midpoint of eac subinterval. Te mes size is given by : max, were : x + x. We define a piecewise polynomial space: V k {v : v P k, }, were P k is te space of polynomials of degree at most k. ote tat te continuity of any function in V k is not required. For v V k, we define v + : lim + v x ɛ ɛ and v : lim + v x ɛ + + ɛ. We use

3 Matematics 8, 6, 3 of 7 {v } + v + + v + and [v + ] + v + v + to denote te average and + ump at x +, respectively. n order to define local discontinuous Galerkin metods, we rewrite Equation 3 using an auxiliary variable q. Terefore, we ave a u, q system given by u tt q x u and q u x. Our local discontinuous Galerkin metods for suc a u, q system can be defined as te following. We look for u and q V k, suc tat u,tt v dx q v x dx + q + v q + v + q w dx u v dx, 4 u w x dx + û + w û + w +, 5 for any v, w V k. Here, û + is te so-called numerical flux wic depends on u + + and u. + Similarly, q + depends on q + + conservation properties, stability and convergence of te numerical scemes. n tis paper, we take te following numerical fluxes û u +, q q. 6 Alternatively, one can also coose û u, q q +... Conservation Properties and q. Te coice of numerical fluxes plays a great role in + n tis subsection, we sow te conservation properties of our semi-discrete scemes 4 and 5. Te energy and linear momentum of te Klein-Gordon equation are given by Ht u t + u x + u dx and Pt u tu x dx, respectively. n te PDE level, te energy and linear momentum of te Klein-Gordon equation are conserved. Our numerical sceme also satisfies te conservation properties. Teorem. Energy Conservation Te numerical solutions u and q to te scemes 4 and 5 wit numerical flux 6 satisfy: d u,t dt + q + u dx. 7 Proof. Let v u,t and q q and obtain in our semi-discrete sceme 4. We ten sum over all te subintervals u,tt u,t dx q u,tx dx q + [u,t ] + u u,t dx. 8 Here, we ave used te periodic boundary condition of q in te equation above. ext, we take a time derivative on bot sides of Equation 5. We ten let w q and û u + and take te sum over all te subintervals. We can obtain q,t q dx u,t q,x dx,t u + [q + ] +. 9

4 Matematics 8, 6, 4 of 7 Here, we ave used te periodic boundary condition of u. We take te sum of Equations 8 and 9 and get te following equation: u,tt u,t + q,t q + u u,t dx q u,tx + u,t q,x dx q [u +,t ] +,t u + [q + ] + [q u,t ] + q [u +,t ] + u,t + [q + ] +. t is easy to see tat te equation above leads to te energy conservation Equation 7, and tis concludes te proof. Teorem sows te energy conserving property of our numerical scemes. A direct conclusion of Teorem is tat u,t + q + u C, were C is a constant tat depends only on te initial data u x,, q x, and u,t x,. Terefore, we ave te following estimates: u C, q C and u,t C, wic implies te stability of our scemes. We ten present te conservation property of linear momentum. Teorem. Linear Momentum Conservation Te numerical solutions u and q to te scemes 4 and 5 wit numerical flux 6 satisfy: d u dt,t q dx [q ] + + [u ] + Proof. Let v q and q q in 4 and sum over all, we ave u,tt q dx q q,x dx q + Let w u in 5 and we sum over all, we can get q u dx Plugging te rigt side of 3 into, we ave u,tt q dx q q,x dx q + [q ] + + [u,t ] +. [q ] + u q dx. u u,x dx u + [u + ] +. 3 [q ] + + u u,x dx + u + [u + ] + [u ] +. 4 We ten take a t derivative on bot sides of Equation 5, sum over all subintervals, and let w u,t, we ave q,t u,t dx u,t u,tx dx,t u + [u +,t ] + By taking te sum of Equations 4 and 5, we can complete te proof. [u,t ] +. 5

5 Matematics 8, 6, 5 of 7 Remark. Teorem sows tat te linear momentum is conserved up to some combination of interior umps of q, u and u,t. n fact, bot te linear momentum and energy can be conserved exactly, if we coose central fluxes in scemes 4 and 5, i.e., q + {q } + and û + {u } + for all. Tis can be proved using te equalities [ f ] { f }[ f ] and [ f g] { f }[g] [ f ]{g} for f, g V k. However, suc a coice of numerical fluxes would lead to sub-optimal orders of convergence. Tat is, wen we use piecewise-defined polynomials of degree k, te L errors of u, q and u t are only of order k, instead of k + as in Teorem 3. n order to obtain te optimal accuracy, we coose alternating fluxes given by Equation Error Estimates n tis subsection, we present te error estimates of our semi-discrete scemes. Let u be te exact solution of te Klein-Gordon 3 and q u x. We denote te error of u and q by e u : u u and e q : q q, respectively, were u and q are te numerical solutions to te semi-discrete scemes 4 and 5 wit numerical flux 6. We ten define te Gauss-Radau proections, denoted by Π and Π +, as follows. For any function v H, te Gauss-Radau proection of v, i.e., Π + v, is a unique function in V k, suc tat Π + vw dx vw dx, w P k ; Π + vx + vx +, 6 for any,,...,. Anoter kind of Gauss-Radau proection, Π v, can be defined similarly. Tat is, Π v, is a unique function in V k, suc tat Π vw dx vw dx, w P k ; Π vx vx, for all. Since te standard L proection will also be needed, we give its definition below. Te L proection of v, denoted by Πv, is a unique function in V k, suc tat Πvw dx vw dx, w P k. 8 Bot Gauss-Radau and standard L proections satisfy te following approximation properties. For any function v H k+, tere exists a positive constant C suc tat v Qv C k+, 9 were Q can be Π +, Π or Π. Here, C depends on v k+ wen Q Π or Π +, and it depends on v k+ wen Q Π. Moreover, C is independent of. Using Gauss-Radau proections, we can rewrite e u ξ u η u, were ξ u u Π + u and η u u Π + u. We also rewrite e q ξ q η q, were ξ q q Π q and η q q Π q. Based on te approximation property 9, we know tat η q and η u C k+, and are independent of te sceme. ow, we are ready to present te error estimates of our semi-discrete scemes. Teorem 3. Error Estimates Let u and q be te numerical solutions to te scemes 4 and 5 wit numerical flux 6, and u and q be te exact solutions to te Klein-Gordon Equation 3. f we take u x, Π + ux, and u,t x, Πu t x,, ten te following error estimates old: u, t u, t C k+, q, t q, t C k+, u t, t u,t, t C k+, were C is a positive constant tat depends on t and exact solution, and is independent of te mes size.

6 Matematics 8, 6, 6 of 7 Proof. We apply numerical flux 6 to Equation 4 and summing over all subinterval, we can obtain u,tt v dx q v x dx q + [v] + u v dx. We ten take te t derivative on bot sides of 5 and summing over all subinterval, we get q,t w dx u,t w x dx,t u + [w] + +. t is easy to sow tat te exact solution u and q also satisfy similar equations: u tt v dx q t w dx Subtracting from, we ave ξ u,tt v dx η u,tt v dx qv x dx q + [v] + uv dx, u t w x dx t u + [w] ξ q v x dx q ξ + [v] + ξ u v dx + η u v dx. 4 ote tat we ave used η qv x dx for any v V k, as well as te fact tat η q qx Π qx, due to te definition of Gauss-Radau proection. Similarly, subtracting 3 from, one can get ξ q,t w dx η q,t w dx ξ u,t w x dx u,t ξ + [w] + +, 5 since η u,tw x dx η u,t +. + Let v ξ u,t in 4 and w ξ q in 5, and summing up tese two equations, we ave ξu,tt ξ u,t + ξ q,t ξ q dx : Λ + Λ + Λ 3, 6 were Λ Λ Λ 3 ηu,tt ξ u,t + η q,t ξ q + η u ξ u,t dx, 7 ξq ξ u,tx + ξ u,t ξ q,x dx ξ q + [ξ u,t ] + u,t ξ + [ξ + q ] +, 8 ξ u ξ u,t dx. 9 Applying Caucy-Scwartz inequality and te approximation property 9 to te formulation of Λ, we can sow tat Λ C k+ ξ u,t + ξ q. n addition, due to te periodic boundary condition, we ave Λ [ξ q ξ u,t ] + q ξ [ξ + u,t ] + u,t ξ + [ξ + q ] +. 3

7 Matematics 8, 6, 7 of 7 Combining te estimates about Λ and Λ, Equation 6 leads to d ξ u,t + ξ q + ξ u C k+ ξ u,t + ξ q C k+ ξ u,t + ξ q + ξ u. dt Terefore, we ave d ξ u,t + ξ q + ξ u C k+. 3 dt We integrate te equation above over time [, t] to obtain ξu,t, t + ξ q, t + ξ u, t ξ u,t, + ξ q, + ξ u, + C k+. 3 ext, we estimate ξ u,t,, ξ q, and ξ u,. Since u x, Π + ux,, we ave ξ u x, u x, Π + ux,, wic leads to ξ u,. n addition, ξ u,t, Πu t, Π + u t, Πu t, Π + u t, u t, Π + u t, C k+. n order to estimate ξ q,, we consider 5 at t and get ξ q, w dx η q, w dx ξ u, w x dx u x, ξ + [w] Here, we ave used te fact tat q, w dx u, w dx ux, + [w] + +. Since ξ u x,, Equation 33 leads to ξ q, w dx η q, w dx for any w V k. Let w ξ q x,, we ave ξ q, η q, ξ q, dx η q, ξ q, C k+ ξ q,, 34 wic leads to ξ q, C k+. Combining all te estimates above, we ave ξ u,t, t + ξ q, t + ξ u, t C k+. 35 Terefore, we obtain te error estimate of u as follows: u, t u, t ξ u, t η u, t ξ u, t + η u, t C k+. Te error estimate of q can be obtained in te same manner. 3. Superconvergence of Local Discontinuous Galerkin Discretization n tis section, we discuss te superconvergence property of our semi-discrete scemes. Before we present te main teorem, we give some important lemmas tat will be used in te proof of te main teorem. Lemma. For any x, let ξ u x, t r t + s x, t x x, were r t is a constant and s x, t P k for any fixed t. Let sx, t V k, and sx, t s x, t, x, for,,...,. Ten, te L norm of te function sx, t satisfies te following inequality Proof. t is easy to sow tat 5 leads to s, t C k+. 36 e q w dx ξ u w x dx + ξ u + w ξ + + u + w +. 37

8 Matematics 8, 6, 8 of 7 Here, we ave used te fact tat η u w x dx and η u + + parts to te term ξ u w x dx, we can rewrite Equation 37 as,. By applying integration by e q w dx ξ u,x w dx + [ξ u ] + w We ten take w s x, tx x + / in 38 to obtain e q w dx x x s x x x + s s x, t s x x + x dx x x + s dx + x x + s s,x dx + s s,x x x + dx. 39 Since x x + s s x x +,x dx s x s,x s dx s dx, we get x x + s s,x dx s x, t s dx. 4 Combining Equations 39 and 4, we ave x x + e q s dx s 4 x, t + s dx. 4 4 Tus, s dx 4 e q s x x + dx. 4 Let dx be a piecewise linear polynomial on, and dx x x + on eac subinterval. t is easy to sow d. Terefore, Equation 4 leads to s sx, t dx 4 e q sx, tdxdx 4 e q s d. 43 We cancel te term s in te inequality above, and applying Teorem 3 to te resulting inequality, we can eventually get s 4 e q d 4 e q C k+. 44 For convenience, let τ be any fixed time, we define Eux, ξ t : τ t ξ u dt, Eqx, ξ t : τ t ξ q dt, Eux, η t : τ t η u dt, Eq η x, t : τ t η q dt, Eux, e t τ t e u dt and Eqx, e t τ t e q dt. Based on tese definitions, we ave te following estimate. Lemma. For any x, let Eq ξ o t + p x, t x x, were o t is a constant and p x, t P k for any fixed t. Letting px, t V k, and px, t p x, t, x, for,,...,. Ten, te L norm of te function px, t satisfies te following inequality: p, t C k+. 45

9 Matematics 8, 6, 9 of 7 Proof. t is easy to sow tat 4 leads to e u tt v dx ξ q v x dx + ξ q v ξ + + q v + e u v dx. 46 Here, we ave used te fact tat η q v x dx and η q + parts to te term ξ q v x dx, we can rewrite Equation 46 as,. By applying integration by e u tt + e u v dx ξ q,x v dx + [ξ q ] v We ten take time integral from t to τ to obtain e u,t x, τ e u,t x, t v dx + Eux, e tv dx Eq ξ x v dx + [Eq] ξ v Let v p x, tx x / in 48, we ave [E ξ q] v + and E ξ q x v dx x x p x x p p x +, t x p x x x dx x x p dx x x p p,x dx p p,x x x dx. 49 Since x x p p x x,x dx p x + p,x p dx p dx, we get x x p p,x dx p x, t p dx. 5 Combining Equations 48 5, we ave x x e u,t x, τ e u,t x, t + Eux, e t p dx p 4 x +, t + p 4 dx. Tus, p dx 4 e u,t x, τ e u,t x, t + Eux, e t p x x dx. 5 Let d x be a piecewise linear polynomial on, and d x x x on eac subinterval. t is easy to sow d. Terefore, Equation 4 leads to p px, t dx 4 e u,t, τ + e u,t, t + Eu, e t p d. 5 One can sow tat Teorem 3 leads to e u,t, τ, e u,t, t, E e u, t C k+. ote tat te constant C in tis inequality also depends on t and τ. We ten cancel te term p in 5, and apply te estimates about e u,t, τ, e u,t, t and E e u, t to te resulting inequality, we can eventually get p C k+ d C k+. 53

10 Matematics 8, 6, of 7 Te next lemma provides some important estimates wic will be used in te proof of superconvergence. Lemma 3. Te following inequalities are satisfied τ τ dt dt η u,t ξ u dx C k+3, 54 E η q ξ q dx C k Proof. We first consider 54. Let d 3 x be a piecewise linear function suc tat d 3 x x x / wen x, for,,...,. Tus, d 3. From Lemma, we obtain τ dt η u,t ξ u dx τ x x dt r + s η u,t dx τ τ dt dt x x s η u,t dx s d 3 η u,t dx τ s η u,t d 3 dt C k Here, we used te fact tat r η u,t dx in te first equality above, and applied Lemma in te last inequality above. Tis completes te proof of 54. ext, we consider τ dt Eη q ξ q dx. Due to te definition of E ξ q, we know d dt Eξ q ξ q and d dt Eη q η q. n addition noting tat E η q x, τ E ξ qx, τ, we tus ave τ dt τ Eq η ξ q dx Eq η τ d dt d dt Eξ qdx Eq η Eq ξ E η q x, E ξ qx, dx τ dx + τ dt Eq η x, p x, x x dx E η q x, px, d 3 xdx E η q, p, d 3 + τ d dt Eη q Eqdx ξ τ η q E ξ qx, tdx τ dt η q p x, t x x dx dt η q px, td 3 xdx η q p, t d 3 dt C k ote tat te last inequality in 57 is based on te estimates about p, and p, t from Lemma, as well as te approximation property about η q and E η q. ow, we are ready to present te main teorem about te superconvergence property of our sceme. Teorem 4. Superconvergence Let u and q be te numerical solutions to te scemes 4 and 5 wit numerical flux 6, and u and q be te exact solutions to te Klein-Gordon Equation 3. f we take u x, Π + ux, and u,t x, Πu t x,, ten te following error estimates old: Π + u, t u, t C k+ 3,

11 Matematics 8, 6, of 7 were C is a positive constant tat depends on t and te exact solution, and is independent from te mes size. Proof. From Equation 4, we get ξ u,tt v dx By rewriting ξ u,ttv dx as d dt ξ u,t v t dx η u,tt v dx η u,tt v dx ξ q v x dx q ξ + [v] + e u v dx. ξ u,tv dx ξ u,tv t dx, te equation above leads to ξ q v x dx q ξ + [v] + e u v dx d ξ u,t v dx. dt We take v E ξ ux, t in te equality above and recall E ξ ux, t τ t ξ u dt wic implies v t ξ u ; tus, we can obtain ξ u,t ξ u dx η u,tt Eudx ξ e u Eudx ξ d ξ u,t Eudx ξ dt From Equation 5, we can sow e q w dx ξ q Eu ξ x dx q ξ [E + u] ξ ξ u w x dx u ξ + [w] We ten take an integral wit respect to t from τ to t on bot sides of Equation 59, and let w ξ q to get Eqξ ξ q dx Eq η ξ q dx E ξ uξ q x dx u E ξ + [ξ + q ] +. 6 ote tat ξ q Eu ξ x dx + q ξ + [Eu] ξ + + Euξ ξ q x dx + u E ξ + [ξ + q ] +. 6 We add Equations 58 and 6 and apply 6 to te resulting equation, and we can sow tat ξ u,t ξ u + Eqξ ξ q + e u Eu ξ dx η u,tt Eudx ξ + Eq η ξ q dx d ξ u,t Eudx ξ dt d e u,t Eudx ξ + η u,t ξ u dx + Eq η ξ q dx. 6 dt Since E ξ qξ q d dt Eξ q and e u E ξ u d dt Eξ u, Equation 6 leads to d ξ u Eq ξ Eu ξ dx η u,t ξ u dx + Eq η ξ q dx d e u,t Eudx. ξ 63 dt dt We take te time integral from to τ in te equation above, and get te following equation ξ u, τ ξ u, + Eq, ξ + Eu, ξ τ τ dt η u,t ξ u dx + dt Eq η ξ q dx + e u,t x, Eux, ξ dx. 64

12 Matematics 8, 6, of 7 Here, we ave used te fact tat Eqx, ξ τ Eux, ξ τ, x. Te first two terms on te rigt side of Equation 64 can be estimated using Lemma 3. Tat is, τ dt η u,tξ u dx + τ dt Eη q ξ q dx C k+3. Since u x, Π + ux,, tere is ξ u,. n addition, u,t x, Πu t x, implies e u,tx, Eux, ξ dx since Eux, ξ V k. To sum up, Equation 64 becomes ξ u, τ + Eq, ξ + Eu, ξ C k Suc an inequality indicates u, τ Π + u, τ ξ u, τ C k+3 for any fixed τ, wic concludes te proof of tis teorem. 4. Time Discretizations n Sections and 3, we ave defined our semi-discrete scemes and presented some conservation properties, error estimates and superconvergence properties. Here, we complete te definition of our fully-discrete scemes by providing two metods of time discretizations. Tat is, we consider explicit and implicit time discretizations for 4 and 5 as follows: Sceme : For n,,..., we look for, q V k, suc tat u n + un δt v dx q w dx for any v, w V k. Sceme : For n,,..., we look for, q V k, suc tat u n + un q δt v dx for any v, w V k. u n v dx, q n v x dx + q n v q n + + v + w x dx + + w w +, + q n v x dx + q + q n q + q n + v + + v + u + u n q w dx w x dx + + w w +, Here, and q represent te numerical solutions at time t : n + δt, were δt is te time step. Te initialization for bot scemes is given as follows: v dx, u Π + ux,, 66 u u + δt Πu tx, + δt u xx u. 67 ote tat Sceme and Sceme are explicit and implicit scemes, respectively. Te property of bot scemes are given in te following teorem.

13 Matematics 8, 6, 3 of 7 Teorem 5. Let u n and qn be te numerical solutions to Sceme wit numerical flux 6 and initialization 66 and 67; ten, te following equality olds u n dx + q q n dx + u n δt u n un dx + q n qn dx + u n un δt. 68 Alternatively, if u n and qn are te numerical solutions to Sceme wit numerical flux 6 and initialization 66 and 67, ten te following equality olds + q + δt u Proof. We first prove te result for Sceme. We rewrite Sceme as u n + un δt v dx q q n δt u n u n + q n + δt un un. 69 q n v x dx q n + w dx δt w x dx [v] + u n v dx, u δt + + [w] +. Let v u equations to get u n δt and w q n in te equations above, and we take te sum of te resulting u n + un δt u δt q q n dx + δt q n dx + u n dx. 7 δt Here, we ave used te equality q n u δt x dx + δt q x dx + + [ ] u q n + δt u + δt + + [q ] + in 7. Since u n + un u n dx + q q n dx + u n δt u n u n un, Equation 7 leads to u n un dx + q n qn dx + ext, we consider te result for Sceme. We rewrite Sceme as u n + un q δt v dx q q n δt w dx + q n + u n δt v x dx v dx, w x dx q u u n un δt + q n [v] + + δt + +. [w] +.

14 Matematics 8, 6, 4 of 7 Let v u u n δt and w q +q n in te equations above, we can furter derive u n + un δt u δt dx + + q q n δt + u n q u + q n dx δt dx, 7 since + q q + q n u δt + q n [ u + x δt dx + ] + + δt u q + q n δt + + x dx [ q ] + q n +. t is easy to sow tat Equation 7 leads to δt u n u n un + q q n + u n, 7 wic is equivalent to 69. Remark. Equation 68 implies te conservation of total energy in te fully-discrete sense. Here, u u n dx is an approximation of u dx, q q n dx is an approximation of q dx, and u u n δt approximates u t dx. 5. umerical Experiments n tis section, we demonstrate te teoretical findings in Sections and 3, including te optimal convergence rates of u u and q q, te superconvergence property of Π + u u, and conservation properties of our sceme, by some numerical tests. We consider te following initial-boundary value problem: u tt u xx + u, x, t [, ], T], 73 ux, sinπx, u t x,, 74 u, t u, t. 75 Te exact solution to te initial-boundary value problem is ux, t sinπx cos 4π + t. We apply sceme defined in Section 4 to solve te problem. Te numerical results by sceme in Section 4 are similar, and tus we skip te discussion. We first estimate te convergence rates of u u, q q and Π + u u. n tese numerical experiments, we use a uniform grid wit / for various. n order to observe te convergence order in space, we coose te time step size δt. so tat te error in space dominates. We compute te numerical solutions at T.5 using P k basis wit k,, 3. To estimate te convergence rate of u u, we take different values of and compute u u. Suppose u u as a convergence rate of r, i.e., u u C r were C is a positive constant; ten, log u u r log + log C. We use several points of log, log u u wit different to fit a straigt line wose slope is an approximation of te convergence rate r. Te convergence rates of q q and Π + u u can be obtained in te same manner. n Figure, we present te error of u and q in L norm wen we use P basis. Te empty circles represent te discrete numerical results and eac red straigt line represents te linear least square fitting of te

15 Matematics 8, 6, 5 of 7 empty circles. Figure a sows tat te convergence rate of u u is approximately.4, wic is consistent wit te optimal convergence rate, i.e., k + wit k, as proved in Section. n addition, te results from Figure b verifies te convergence rate of for q q. As for te convergence rate of Π + u u, it is actually greater tan.5. Similarly, from Figures and 3, we can observe te convergence rates of k + for u u and q q wen P and P 3 basis are used. Figure c sows tat Π + u u as te convergence rate of approximately 4 wen we use P basis, wic indicates te k + t order of convergence wit k. Figure 3c gives te convergence rate for te case of P 3 basis. We can see tat Π + u u is convergent at te rate of , wic is a little over k + 3/ wit k 3. Based on te numerical results in Figures 3, as well as te discussion above, we can draw te conclusion tat our numerical results coincide wit our rigorous proof about error estimates and te superconvergence property log a log u u versus log. Slope log b log q q versus log. Slope log c log Π + u u versus log. Slope Figure. log E versus log, were E is te L error at T.5 wit P basis log a log u u versus log. Slope log b log q q versus log. Slope log c log Π + u u versus log. Slope Figure. log E versus log, were E is te L error at T.5 wit P basis log a log u u versus log. Slope log b log q q versus log. Slope log c log Π + u u versus log. Slope Figure 3. log E versus log, were E is te L error at T.5 wit P 3 basis. ext, we ceck te conservation properties of our scemes using a simulation over a long time. As indicated by Teorems and in Section., and Teorem 5 in Section 4, te conservation

16 Matematics 8, 6, 6 of 7 properties of energy and linear momentum are independent of te degree of polynomials. Here, we only present te long time simulation wen P basis is used. We still consider te initial-boundary value problem We coose., δt 4 and run te simulation up to T. Using te exact solution ux, t sinπx cos 4π + t, we can sow tat te linear momentum Pt u tu x dx for all t. umerically, we approximate te linear momentum using te discrete form of Pt. Tat is, we compute u n te time istory of te error in linear momentum in Figure 4b. /δt q dx for n,,..., 6, and present We can observe tat te error oscillates from T to T, and te magnitude of oscillations increase up to T 76 and ten decrease from T 76 all te way to T. Overall, te linear momentum is conserved up to te magnitude of. Teorem implies tat te rate of cange for linear momentum depends on some umps at interior interfaces, and te semi-discrete form of linear momentum is not conserved up to macine epsilon. However, our numerical results indicate tat te quantity [q ] + + [u ] + [u,t] +, altoug not equal to zero analytically, is not far away from macine epsilon numerically. Terefore, we can observe te conservation of linear momentum numerically. As for te conservation of total energy, we compute te discrete form of energy using u u n dx + q q n dx + u n /δt for n,,..., 6. Figure 4a sows te error in te total energy. We can observe tat te total energy is conserved up to te magnitude of 9. Terefore, we ave verified te energy conservation and linear momentum conservation troug te numerical experiments T a Error in te total energy T b Error in te momentum Figure 4. Time istory of te error in te total energy and linear momentum wit P basis and.. left error in te total energy; rigt error in te linear momentum. 6. Conclusions n tis paper, ig-order energy and linear momentum conserving metods are proposed for te Klein-Gordon equation. Our numerical metods are based on te local discontinuous Galerkin metods. By coosing alternating numerical fluxes 6, we can prove tat te total energy of te equation is exactly conserved and te linear momentum is conserved up to some terms at element interface. Results from long time numerical simulations up to T indicate tat te linear momentum is conserved up to te magnitude of. Moreover, suc a coice of numerical fluxes leads to optimal convergence order of u, q and u,t, and te superconvergence property. t is important to mention tat te scemes proposed in tis paper can be directly generalized to te Klein-Gordon equation wit external potential. Moreover, te results from tis paper sed ligt on te design of ig-order structure-preserving metods for nonlinear coupling systems involving te Klein-Gordon equation, for example, te Klein-Gordon-Scrödinger equations and te Klein-Gordon-Zakarov equations, wic is currently under investigation. Acknowledgments: Te autor acknowledges te support from Augusta University. Te autor would also like to tank tree anonymous reviewers for teir insigtful and constructive comments, wic elp to improve te quality of tis paper. Conflicts of nterest: Te autor declares no conflict of interest.

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