and Chi-Wang Shu

https://doi.org/10.1051/m2an/2018069 www.esaim-m2an.org

STABILITY ANALYSIS AND ERROR ESTIMATES OF ARBITRARY

LAGRANGIAN–EULERIAN DISCONTINUOUS GALERKIN METHOD COUPLED WITH RUNGE–KUTTA TIME-MARCHING FOR LINEAR

CONSERVATION LAWS

Lingling Zhou

, Yinhua Xia

and Chi-Wang Shu

Abstract.In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needsτ ≤ρh² and the second order TVD-RK scheme needsτ ≤ρh⁴³ for higher order polynomials in space, whereτ andhare the time and maximum space step, respectively, andρis a positive constant independent ofτ andh.

Mathematics Subject Classification. 65M60, 35L65, 65M12.

Received March 8, 2018. Accepted November 2, 2018.

1. Introduction

In this paper, we consider the stability analysis and error estimates of an arbitrary Lagrangian–Eulerian dis- continuous Galerkin (ALE-DG) method coupled with Runge–Kutta time-marching schemes for one-dimensional linear conservation laws

ut+ (βu)x= 0, (x, t)∈[a, b]×(0, T],

u(x,0) =u₀(x), x∈[a, b] (1.1)

Keywords and phrases.Arbitrary Lagrangian–Eulerian discontinuous Galerkin method, Runge–Kutta methods, stability, error estimates, conservation laws.

1 School of Mathematics and Information Science, Henan Polytechnic University, 454003 Jiaozuo, Henan, PR China.

2 School of Mathematical Sciences, University of Science and Technology of China, 230026 Hefei, Anhui, PR China.

3 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.

∗Corresponding author:yhxia@ustc.edu.cn

Article published by EDP Sciences c EDP Sciences, SMAI 2019

with the periodic boundary condition. Here β is a constant. We only pay attention to the smooth solution of (1.1).

The discontinuous Galerkin (DG) method is a class of finite element methods, in which the basis functions are completely discontinuous, piecewise polynomials. Reed and Hill introduced the first DG method to solve the neutron equation [19] and later, Cockburn et al. extended the method to Runge–Kutta DG (RKDG) for nonlinear conservation laws in a series of papers [2,4–6]. The DG method has a wide range of applications owing to some advantages like parallelization capability, the strong stability and high-order accuracy, and so on. We refer to [3,7,8,12,20] and the references therein for more references of the DG method.

For the theoretical analysis of the fully discrete DG method, Zhanget al.have done a lot of work for conserva- tion laws [22–26], where the time discretization is the explicit second or third order total variation diminishing Runge–Kutta (TVD-RK) method. For the smooth solutions, they obtained (quasi)-optimal error estimates for both the second and third order TVD-RK time-marching schemes with periodic boundary conditions and suitable CFL conditions. The stability of the third order TVD-RK (TVD-RK3) was shown in [25]. They also considered the inflow boundary condition as well as the discontinuous initial data [22,26]. Moreover, Burmanet al.[10] analyzed the explicit RK schemes in combination with stabilized finite element methods for first-order linear partial differential equation systems and established sub-optimal error estimates for smooth solutions, which presented a unified analysis for several high-order symmetrically stabilized finite element methods en- countered in the literature. We refer to [15,21] for the energy analysis, which is the main technique for all work listed above.

However, all the analysis listed above are considered on the static grids. The ALE-DG method discussed here is a moving mesh DG method and the grid moving methodology belongs to the class of arbitrary Lagrangian–

Eulerian (ALE) methods [9], which allows the motion of the mesh to be like either the Lagrangian or the Eulerian description of motion and should satisfy the geometric conservation law (GCL). The significance of the GCL has been analyzed by Guillard and Farhat [11]. There have been works about the implementation and applications of the ALE-DG method in the literature,e.g., [14,16–18]. Klingenberget al. developed an ALE-DG method for one-dimensional conservation laws [13], where local affine linear mappings connecting the cells for the current and next time level are defined and yield the time-dependent approximation space. They showed that the ALE-DG method satisfies the GCL for any Runge–Kutta scheme and is efficient for the conservation law. They also showed that the ALE-DG method shares many good properties of the DG method defined on static grids,e.g., theL²stability, the local maximum principle, high order accuracy, and so on.

The main purpose of our work is to study the stability and the error estimates for the ALE-DG method combined with the explicit Runge–Kutta time-marching schemes, in which the Euler-forward, the second order TVD-RK (TVD-RK2) and TVD-RK3 methods are considered. Compared with the work on the static grids, our analysis is similar but more technical. Owing to the time-dependent functional space, the scaling arguments play an important role in this work. With the energy estimates, we prove that all three fully discrete schemes are stable under suitable CFL conditions. More precisely, for the Euler-forward scheme with P⁰ (piecewise constant) elements, the TVD-RK2 scheme with P¹ (piecewise linear) elements and the TVD-RK3 approach with polynomials of any order in space, the usual CFL condition is needed, while the Euler-forward scheme with P^k elements for k≥1 requires τ ≤ρh² and the TVD-RK2 approach with P^k elements for k≥2 needs τ ≤ρh⁴³ for the results to hold true. Hereτ andhare the time and maximum spatial mesh sizes, respectively, andρis a positive constant independent ofτandh. To best understand the error equations, we reformulate the equation (1.1) in terms of a suitable coordinate transformation. Then we proceed to obtain quasi-optimal error estimates in space and optimal convergence rates in time under the same CFL condition as the stability. To the best of our knowledge, the above results are the first for high order ALE methods with minimum smoothness assumptions on mesh movements (only assuming uniform Lipschitz continuity of the mesh movements) and without the need of remapping.

The organization of our paper is as follows. In Section 2, we list some notations adopted throughout the paper. The semi-discrete ALE-DG scheme for the linear conservation law is given in Section 3, where we also show some properties of the scheme. Section 4 presents the stability of the ALE-DG scheme in combination

with the explicit RK time-marching methods up to third order. The error estimates for the three corresponding fully discrete schemes are proven in Section5. We conclude our results in Section6.

2. Notations

In this section, we will introduce some notations adopted throughout the paper.

2.1. Notations for the distribution of the mesh

Let Ω = [a, b]. In order to describe the semi-discrete ALE-DG scheme of equation (1.1), we first introduce some notations for the distribution of the mesh. Assume that the mesh generating pointsn

xⁿ_j−1 2

oN j=1

are given at any time level tn, n= 0, . . . , M, and the points xⁿ_j−1

2

and xⁿ⁺¹

j−¹₂ are connected by time-dependent straight lines

x_j−1

2(t) :=xⁿ_j−1 2

+ω_j−1

2(t−t_n), ∀t∈[t_n, t_n+1], (2.1) where

ω_j−1 2 :=

xⁿ⁺¹

j−¹₂ −xⁿ

j−¹₂

tn+1−tn

. (2.2)

Note that for any time t, the first point x1

2(t) and the last point x_N+1

2(t) stay the same for compactly supported problems and could move with the same speed _dt^dx1

2(t) =_dt^dx_N+1

2(t) for periodic boundary problems.

We provide an example to show the distribution of the ALE mesh in Figure1. The straight lines (2.1) provide the time-dependent cells

Kj(t) := [x_j−1

2(t), x_j+1

2(t)], ∀t∈[tn, tn+1] and j= 1, . . . , N.

The length of each cellKj(t) is denoted by ∆j(t) :=x_j+1

2(t)−x_j−1

2(t). Moreover, we seth(t) := max

1≤j≤N∆j(t) andh:= max

t∈[0,T]h(t). We assume that the mesh is quasi-uniform in the sense thath≤C∆j(t) forj = 1,2, . . . , N,

a b

b xj+1/2

x_j-1/2ⁿ n

x_j-1/2ⁿ⁺¹ x_j+1/2ⁿ⁺¹

t_n t_n+1

a

Figure 1. An example of the ALE mesh.

whereCis a positive constant and independent ofh. In addition, the grid velocity field for allt∈[t_n, t_n+1] and x∈Kj(t) is defined by

ω(x, t) =ω_j+1 2

x−x_j−1 2(t)

∆j(t) +ω_j−1 2

x_j+1 2(t)−x

∆j(t) , (2.3)

and the weak derivative ofω with respect toxis given by

∂x(ω(x, t)) = ω_j+1

2 −ω_j−1 2

∆j(t) = ∆⁰_j(t)

∆j(t), (x, t)∈Kj(t)×[tn, tn+1]. (2.4) Note that∂_x(ω(x, t)) is independent ofx. The quasi-uniformity assumption for the meshes implies that ∆_j(t) satisfies the following property, for allt∈[tn, tn+1],n= 0, . . . , M−1,

∆j(t) = (ω_j+¹

2 −ω_j−¹

2)(t−tn) + ∆j(tn)>0. (2.5) In addition, we assume thatω(x, t) satisfies the following properties:

(ω1): There exists a constantC_w≥0, independent of h, such that, max

(x,t)∈[a,b]×[0,T]|ω(x, t)| ≤C_w; (2.6)

(ω2): There exists a constantCwx≥0, independent ofh, such that, max

(x,t)∈[a,b]×[0,T]|∂x(ω(x, t))| ≤Cwx. (2.7) For anyKj(t), we define the following time-dependent linear mapping

χj: [−1,1]−→Kj(t), χj(ξ, t) =∆_j(t)

2 (ξ+ 1) +x_j−¹

2(t), (2.8)

which yields a characterization of the grid velocity

∂_t(χ_j(ξ, t)) =ω(χ_j(ξ, t), t), ∀(ξ, t)∈[−1,1]×[t_n, t_n+1].

For simplicity, we denoteK_jⁿ≡K_j(t_n) and ∆ⁿ_j ≡∆_j(t_n), for anyn= 1, . . . , M.

2.2. Notations for function space and norms For anyt∈[tn, tn+1], the finite element space is defined by

Vh(t) :={v∈L²(Ω) :v(χj(·, t))∈P^k([−1,1]), j= 1,2, . . . , N},

whereP^k([−1,1]) denotes the space of polynomials of degree at mostkon [−1,1]. We denote the inner product over the intervalKj(t) and the associated norm by

(v, r)_Kj(t)= Z

Kj(t)

vrdx, kvk_Kj(t)=q

(v, v)_Kj(t).

We also use the usual notations of Sobolev space. Let H^s(D) be the Sobolev space on sub-domain D⊂Ω, which is equipped with the normk · k_Hs(D) for any integers≥0. Then we define the broken Sobolev space

H_h¹(t) :={v:v(χ_j(·, t))∈H¹([−1,1]), j= 1,2, . . . , N},

which contains the finite element space. Moreover, the left and right limits ofvat the pointx_j−1

2(t) are denoted byv⁻

j−¹₂ andv⁺

j−¹₂, respectively, where v^±

j−¹₂ = lim

ε→0⁺v(x_j−1

2(t)±ε, t).

Thus the cell average and the jump at the pointx_j−¹

2(t) are defined by {{v}}_j−1

2 = 1 2

v⁺

j−¹₂ +v⁻

j−¹₂

, [[v]]_j−1 2 =v⁺

j−¹₂ −v⁻

j−¹₂. Summing over all the elements, we denote

(v, r) =

N

X

j=1

(v, r)_Kj(t), kvk²=

N

X

j=1

kvk²_K

j(t), [[v]]²=

N

X

j=1

[[v]]²_j−1 2

.

LetΓ_h(t) be the union of all elements interface points and define theL²-norm onΓ_h(t) by

kvkΓ_h(t)=





N

X

j=1

|v_j−⁺ 1 2

|²+|v⁻_j+1 2

|²





1/2

.

2.3. Notations for coordinate transformation

In the following, we will introduce some notations for coordinate transformations, which are often used in our stability analysis. For simplicity, we only consider the uniform partition of the time interval [0, T], namely {tn =nτ}^M_n=0 with the time stepτ andM τ =T. With the time-dependent linear mapping (2.8), we have, for three different time stagest_n,t_n+1

2 =t_n+^τ₂, andt_n+1,

[−1,1]7−→K_jⁿ, χj(ξ, tn) =∆ⁿ_j

2 (ξ+ 1) +xⁿ_j−1 2

, (2.9)

[−1,1]7−→Kⁿ⁺

1 2

j , χj(ξ, t_n+1

2) =∆ⁿ⁺

1 2

j

2 (ξ+ 1) +xⁿ⁺

1 2

j−¹₂, (2.10) [−1,1]7−→K_jⁿ⁺¹, χ_j(ξ, t_n+1) =∆ⁿ⁺¹_j

2 (ξ+ 1) +xⁿ⁺¹

j−¹₂. (2.11) Thus∀φ∈Vh(tn),ϕ∈Vh(t_n+1

2), andψ∈Vh(tn+1), define φˆ

χj(·, tn+1)

:=φ

χj(·, tn)

, φ¯

χj(·, t_n+¹

2)

:=φ

χj(·, tn)

, (2.12)

ˆ ϕ

χ_j(·, tn+1)

:=ϕ

χ_j(·, t_n+¹

2)

, ϕ˜

χ_j(·, tn)

:=ϕ

χ_j(·, t_n+¹

2)

, (2.13)

ψ˜

χ_j(·, t_n)

:=ψ

χ_j(·, t_n+1)

, ψ¯

χ_j(·, t_n+1 2)

:=ψ

χ_j(·, t_n+1)

. (2.14)

Moreover, from (2.4) and (2.5), we get

∆ⁿ_j

∆ⁿ⁺¹_j = 1−s₂>0, ∆ⁿ⁺¹_j

∆ⁿ_j = 1 +s₁>0, (2.15)

∆ⁿ_j

∆ⁿ⁺

1 2

j

= 1−s₃

2 >0, ∆ⁿ⁺

1 2

j

∆ⁿ_j = 1 +s₁

2 >0, (2.16)

∆ⁿ⁺_j ¹²

∆ⁿ⁺¹_j = 1−s₂

2 >0, ∆ⁿ⁺¹_j

∆ⁿ⁺_j ¹²

= 1 +s₃

2 >0, (2.17)

where

s1=τ ωx(tn), s2=τ ωx(tn+1), s3=τ ωx(t_n+1

2), (2.18)

andω_x(t)≡∂_xω(x, t) is given by (2.4). Note that

∆ⁿ_j

∆ⁿ⁺¹_j ·∆ⁿ⁺¹_j

∆ⁿ_j = 1, ∆ⁿ_j

∆ⁿ⁺

1 2

j

·∆ⁿ⁺

1 2

j

∆ⁿ_j = 1, ∆ⁿ⁺

1 2

j

∆ⁿ⁺¹_j · ∆ⁿ⁺¹_j

∆ⁿ⁺

1 2

j

= 1, we have

s1=s2+s1s2, s1=s3+s1s3

2 , s3=s2+s2s3

2 · (2.19)

In the end, we present some properties. For any functionvⁿ_h∈Vh(tn), the scaling arguments and the assump- tion (2.7) ofωxindicate that,

kv_hⁿk²_Kn j = ∆ⁿ_j

∆ⁿ⁺¹_j kcv_hⁿk²_Kn+1 j

= (1−s2)kcvⁿ_hk²_Kn+1 j

≤(1 +Cwxτ)kcv_hⁿk²_Kn+1 j

, (2.20)

kcvⁿ_hk²_Kn+1 j

=∆ⁿ⁺¹_j

∆ⁿ_j kv_hⁿk²_Kn

j = (1 +s1)kv_hⁿk²_Kn

j ≤(1 +Cwxτ)kv_hⁿk²_Kn

j. (2.21)

Similarly, we also have kvⁿ_hk²

K^{n+ 1}_j ²

≤(1 +Cwx

2 τ)kcvⁿ_hk²_Kn+1 j

, kcvⁿ_hk²_Kn+1 j

≤(1 +Cwx

2 τ)kvⁿ_hk²

K_j^{n+ 12}

. (2.22)

Remark 2.1. The introduction of coordinate transformations (2.9)–(2.11) is to make the presentations simpli- fied and clear. With the help of them, the relation betweenφ, ˆφand ¯φin (2.12), the representations of the same function at different time stages, is easy to be understood. Moreover, it is straightforward to obtain properties (2.20)–(2.22), which are frequently used in our analysis.

2.4. Projections and inverse properties

In this section, we will present two types of projections. TheL²projectionPhand Gauss-Radau projections P_h^±intoVh(t), which are often used to derive the quasi-optimal and optimalL²error bounds of the DG method.

For a functionu∈L²(Ω), theL² projection is defined by

(Phu, v)_Kj(t)= (u, v)_Kj(t), ∀v∈Vh(t). (2.23)

Fork≥1 andv(χ(·, t))∈P^k−1([−1,1]), the Gauss-Radau projections are defined by (P_h⁻u, v)_Kj(t)= (u, v)_Kj(t), P_h⁻u

x⁻

j+¹₂(t)

=u x⁻

j+¹₂(t) , (P_h⁺u, v)_Kj(t)= (u, v)_Kj(t), P_h⁺u

x⁺

j−¹₂(t)

=u x⁺

j−¹₂(t) .

(2.24) Let Qhube either Phuor P_h^±u. Suppose u∈H^k+1(Ω), then by a standard scaling argument, it is easy to show (c.f.[1]) for both projections that

kηk+h^1/2kηk_Γh(t)+hk∂xηk ≤Ch^k+1, (2.25) where η =Qhu−uand the positive constantC depends on uand its derivatives, but it is independent of h.

Finally, we present the well-known inverse properties of the finite element spaceVh(t). For anyv∈Vh(t), there exists positive constantsµ1and µ2, independent ofv andh, such that

hkvxk ≤µ1kvk, h¹²kvkΓ_h(t)≤µ2kvk. (2.26) In the following, we denote µ= max{µ1, µ²₂}. For more details of the inverse property, we refer the reader to [1].

3. Semi-discrete ALE-DG method

To derive the semi-discrete ALE-DG method, we first list the following lemma, which has been proven in [13].

Lemma 3.1. Let ube a sufficiently smooth function in any cell K_j(t). Then for allv ∈V_h(t), there holds the transport equation

d

dt(u, v)_Kj(t)= (∂tu, v)_Kj(t)+ (∂x(ωu), v)_Kj(t), ∀j= 1, . . . , N. (3.1) Next, multiply the equation (1.1) by a test functionv∈Vh(t) and apply the integration by parts as well as the transport equation (3.1), we obtain the semi-discrete ALE-DG method for arbitraryKj(t),t∈[tn, tn+1]: find uh∈Vh(t) such that for all test functionsv∈Vh(t), we have

d

dt(u_h, v)_Kj(t)= (g(ω, u_h), v_x)_Kj(t)−g(ω, uˆ _h)_j+1 2v⁻_j+1

2

+ ˆg(ω, u_h)_j−1 2v⁺

j−¹₂, (3.2)

where g(ω, uh) = (β−ω)uh and the numerical flux ˆg(ω, uh)_j−1

2 can be chosen as the Lax-Friedrichs flux, for j= 1, . . . , N,

ˆ

g(ω, u_h)_j−1

2 = (β−ω_j−1

2){{uh}}_j−¹

2 −α

2[[u_h]]_j−1

2, α= max

Ω×[tn,tn+1]|β−ω|. (3.3) For simplicity, we define the ALE-DG spatial operatorAas

A(v, r)(t) =

N

X

j=1

A(v, r)K_j(t), ∀v, r∈H_h¹(t), (3.4) where

A(v, r)_Kj(t)=−

(β−ω)v, rx

Kj(t)

+ ˆg(ω, v)_j+1 2r_j+⁻ 1

2

−ˆg(ω, v)_j−1 2r_j−⁺ 1

2

, (3.5)

and ˆg(ω, v)_j−1

2 is the Lax-Friedrichs flux defined by (3.3). Then by the above notations, the semi-discrete ALE-DG scheme (3.2) can be rewritten as

d

dt(u_h, v)_Kj(t)=−A(u_h, v)_Kj(t), ∀v∈V_h(t).

3.2. The properties of the ALE-DG scheme

In this subsection, we shall present some properties of the operator A defined by (3.4), which implies the properties of the ALE-DG spatial discretization.

Lemma 3.2 (Boundedness of the operatorA). SupposeA is defined by (3.4), then for any v, r∈ Vh(t) and t∈[t_n, t_n+1], we have

|A(v, r)(t)| ≤3αµh⁻¹kvkkrk, (3.6)

|A(v, r)(t)| ≤

αkv_xk+C_wxkvk+√

2αµh⁻¹²[[v]]

krk. (3.7)

Moreover, for the piecewise linear case, i.e., k= 1 in the finite element spaceV_h(t), there holds

|A(v, r−P_h⁰r)(t)| ≤

(µ+ 1)C_wxkvk+√

2αµh⁻¹²[[v]]

kr−P_h⁰rk, (3.8)

whereP_h⁰rdenotes the L² projection ofronto the piecewise constant finite element space.

Proof. By the periodic boundary condition, we first obtain A(v, r)(t) =−

(β−ω)v, rx

−

N

X

j=1

ˆ

g(ω, v)_j+¹

2[[r]]_j+¹

2. (3.9)

The definition (3.3) yields,

|ˆg(ω, v)_j+1

2| ≤α(|v⁺_j+1 2

|+|v⁻_j+1 2

|).

Then sum over allj to get

N

X

j=1

ˆ

g(ω, v)²_j+1 2

≤2

N

X

j=1

α²

|v⁺_j+1 2

|²+|v⁻_j+1 2

|²

= 2α²kvk²_Γ

h(t). (3.10)

In addition, we have the following estimates

N

X

j=1

[[r]]²_j+1 2

≤2

N

X

j=1

|r⁺_j+1 2

|²+|r⁻_j+1 2

|²

= 2krk²_Γ

h(t), (3.11)

N

X

j=1

{{r}}²_j+1 2 ≤ 1

2

N

X

j=1

|r_j+⁺ 1 2

|²+|r⁻_j+1 2

|²

=1

2krk²_Γh(t). (3.12)

Thus we can obtain the first inequality (3.6),

|A(v, r)(t)| ≤αkvkkrxk+ ^N

X

j=1

ˆ

g(ω, v)²_j+1 2

¹₂^N X

j=1

[[r]]²_j+1 2

¹₂

≤αµ₁h⁻¹kvkkrk+ 2αkvk_Γh(t)krk_Γh(t)

≤3αµh⁻¹kvkkrk.

Here we use the Cauchy–Schwarz inequality as well as the inverse property (2.26). To obtain the second inequality (3.7), we integrate (3.5) by parts and sum over allj,

A(v, r)(t) =

(β−ω)vx, r

+

∂x(β−ω)v, r

+

N

X

j=1

(β−ω_j+1 2)[[v]]_j+1

2{{r}}_j+1 2 +

N

X

j=1

α 2[[v]]_j+1

2[[r]]_j+1 2

=−ωx(t)(v, r) +B(v, r)(t),

(3.13)

where

B(v, r)(t) =

(β−ω)v_x, r

+

N

X

j=1

(β−ω_j+1

2)[[v]]_j+1

2{{r}}_j+1 2 +

N

X

j=1

α 2[[v]]_j+1

2[[r]]_j+1 2.

Here we use the fact that the quantity ω_x(t) defined by (2.4) only depends ont. By (3.12) and the similar arguments to estimate (3.6), we get

|B(v, r)(t)| ≤αkvxkkrk+√

2α[[v]]krk_Γh(t)

≤

αkvxk+√

2αµh⁻¹²[[v]]

krk, (3.14)

which yields the desired result (3.7),

|A(v, r)(t)| ≤

αkvxk+Cwxkvk+√

2αµh⁻¹²[[v]]

krk.

Here we use the property (2.7) of∂_xω(x, t). Finally, we analyze the inequality (3.8). By the property of the piecewise constantL² projection,

(r−P_h⁰r, vx)_Kj(t)= 0, ∀v(χj(·, t))∈P¹([−1,1]), we have

(β−ω)vx, r−P_h⁰r

K_j(t)

=

(ω_j−1

2 −ω)vx, r−P_h⁰r

K_j(t)

, which yields

(β−ω)v_x, r−P_h⁰r

≤C_wxhkv_xkkr−P_h⁰rk

≤µC_wxkvkkr−P_h⁰rk.

Henceforth, replacingrwithr−P_h⁰rin (3.13) and by similar arguments, we obtain

|A(v, r−P_h⁰r)(t)| ≤

(µ+ 1)Cwxkvk+√

2αµh⁻¹²[[v]]

kr−P_h⁰rk.

Lemma 3.3. SupposeAis defined by (3.4) andP_hv is theL²projection defined by (2.23). Denoteη=P_hv−v, then for any v∈H_h¹(t),r∈Vh(t)andt∈[tn, tn+1], we have

|A(η, r)(t)| ≤µCwxkηkkrk+√

2αkηk_Γh(t)[[r]], (3.15)

|A(η, r)(t)| ≤µCwxkηkkrk+ 2αµh⁻¹²kηk_Γh(t)krk. (3.16)

Proof. From (3.9) and the definition of theL² projection (2.23), we have A(η, r)(t) =−

(β−ω)η, rx

−

N

X

j=1

ˆ

g(ω, η)_j+1 2[[r]]_j+1

2

=

(ω−ω_j−1 2)η, r_x

−

N

X

j=1

ˆ

g(ω, η)_j+1 2[[r]]_j+1

2, which implies that

|A(η, r)(t)| ≤Cwxhkηkkrxk+ ^N

X

j=1

ˆ

g(ω, η)²_j+1 2

¹₂ [[r]]

≤µC_wxkηkkrk+√

2αkηkΓ_h(t)[[r]].

Here we use the Cauchy–Schwarz inequality, the inverse inequality (2.26) as well as the estimate (3.10). It is easy to obtain (3.16) from (3.15) by using the estimate (3.11) and the inverse property (2.26).

Lemma 3.4. For any v,r∈H_h¹(t) andt∈[t_n, t_n+1], we have A(v, r)(t) +A(r, v)(t) =

N

X

j=1

α[[v]]_j+1 2[[r]]_j+1

2 −

N

X

j=1

ω_x(t)(v, r)_Kj(t), (3.17)

A(v, v)(t) =

N

X

j=1

α 2[[v]]²_j+1

2

−

N

X

j=1

ωx(t) 2 kvk²_K

j(t). (3.18)

Proof. With the representation ofA(v, r)(t) in (3.9) and integration by parts, we can easily obtain A(v, r)(t) +A(r, v)(t) =−

(β−ω)v, r_x

−

N

X

j=1

ˆ

g(ω, v)_j+1 2[[r]]_j+1

2

−

(β−ω)r, vx

−

N

X

j=1

ˆ

g(ω, r)_j+1 2[[v]]_j+1

2

=−

N

X

j=1

ωx(t)(v, r)_Kj(t)+

N

X

j=1

α[[v]]_j+1 2[[r]]_j+1

2

−

N

X

j=1

(β−ω_j+¹

2)v⁻_j+1 2

r_j+⁻ 1 2

+

N

X

j=1

(β−ω_j−¹

2)v_j−⁺ 1 2

r_j−⁺ 1 2

−

N

X

j=1

(β−ω_j+1

2)({{v}}_j+1 2[[r]]_j+1

2 +{{r}}_j+1 2[[v]]_j+1

2)

=−

N

X

j=1

ωx(t)(v, r)K_j(t)+

N

X

j=1

α[[v]]_j+¹

2[[r]]_j+¹

2.

Here in the last step we use the periodic boundary condition and [[r]]{{v}}+ [[v]]{{r}}= [[rv]]. It is clear that

(3.17) implies (3.18) ifr=v.

It is worth pointing out that the properties of Ain Lemmas 3.2–3.4 are similar to those in Zhang and Shu [25] developed for the static grids, which play very important roles in obtaining stability.

4. Stability analysis for linear conservation laws

In this section, we would like to analyze the stability of three fully discrete schemes, that is, the ALE-DG method coupled with Euler-forward, TVD-RK2 and TVD-RK3 time-marching schemes. In what follows, we denote the approximation ofu_h(t_n) byuⁿ_h.

4.1. First order scheme

The ALE-DG with the Euler-forward scheme is given in the following form: finduⁿ⁺¹_h ∈Vh(tn+1), such that for anyv_hⁿ≡vh(·, tn)∈Vh(tn) and 1≤j≤N, there holds

(uⁿ⁺¹_h ,cv_hⁿ)_Kn+1 j

= (uⁿ_h, vⁿ_h)_Kⁿ

j −τA(uⁿ_h, vⁿ_h)_Kⁿ

j. (4.1)

Herevcⁿ_h is defined by (2.12).

Theorem 4.1. Let uⁿ⁺¹_h be the numerical solution of the fully discrete scheme (4.1), then we have for anyn, that

kuⁿ⁺¹_h k²≤(1 +Cτ)kuⁿ_hk², under the CFL condition

α²µ²τ h⁻²≤ 1

9· (4.2)

In particular, for the piecewise constant finite element space,Vh(t) ={v(χj(·, t))∈P⁰([−1,1])}, we have the strong stability

kuⁿ⁺¹_h k ≤ kuⁿ_hk, with the usual CFL condition

αµ²τ h⁻¹≤ 1

4· (4.3)

Here αis defined by (3.3),µis the inverse constant (2.26), andC is a positive constant depending solely on C_wx.

Proof. To analyze the stability of the scheme (4.1), we need first to obtain the energy identity. Takevⁿ_h=uⁿ_hin the scheme (4.1) to yield

(uⁿ⁺¹_h ,ucⁿ_h)_Kn+1

j =kuⁿ_hk²_Kn

j −τA(uⁿ_h, uⁿ_h)K_jⁿ. (4.4) By the scaling argument with (2.15), we have

kucⁿ_hk²_Kn+1 j

=∆ⁿ⁺¹_j

∆ⁿ_j kuⁿ_hk²_Kn

j = (1 +s1)kuⁿ_hk²_Kn

j. (4.5)

Noting that

(uⁿ⁺¹_h ,ucⁿ_h)_Kn+1

j = 1

2kuⁿ⁺¹_h k²_Kn+1 j

+1

2kcuⁿ_hk²_Kn+1 j

−1

2kuⁿ⁺¹_h −ucⁿ_hk²_Kn+1 j

,

we get the energy identity by summing up (4.4) overj, 1

2kuⁿ⁺¹_h k²−1

2kuⁿ_hk²= 1

2kuⁿ⁺¹_h −ucⁿ_hk²−

N

X

j=1

s1

2kuⁿ_hk²_Kn

j −τA(uⁿ_h, uⁿ_h)(tn)

= 1

2kuⁿ⁺¹_h −ucⁿ_hk²−τ

2α[[uⁿ_h]]². (4.6)

Here in the last step we use the property (3.18) ofAas well as the definition (2.18) ofs1. Next, we only need to analyze the first term of the right hand side in (4.6). Apply the scaling arguments and (2.15) again to get

(uⁿ_h, v_hⁿ)K_jⁿ= ∆ⁿ_j

∆ⁿ⁺¹_j (cuⁿ_h,vc_hⁿ)_Kn+1

j = (1−s2)(cuⁿ_h,vcⁿ_h)_Kn+1

j , (4.7)

and

A(uⁿ_h, vⁿ_h)K_jⁿ=A(cuⁿ_h,vc_hⁿ)_Kn+1

j . (4.8)

It implies the equivalent form of (4.1), (uⁿ⁺¹_h −ucⁿ_h,cv_hⁿ)_Kn+1

j =−s2(cuⁿ_h,cv_hⁿ)_Kn+1

j −τA(cuⁿ_h,cvⁿ_h)_Kn+1

j . (4.9)

P^k case. Take the test functioncv_hⁿ=uⁿ⁺¹_h −ucⁿ_h in (4.9) and sum all overj to obtain kuⁿ⁺¹_h −ucⁿ_hk²=−

N

X

j=1

s2(cuⁿ_h, uⁿ⁺¹_h −ucⁿ_h)_Kn+1

j −τA(cuⁿ_h, uⁿ⁺¹_h −cuⁿ_h)(tn+1). (4.10) Using the Cauchy–Schwarz inequality and the boundedness (3.6) of the operatorA, we have

kuⁿ⁺¹_h −ucⁿ_hk²≤(Cwxτ+ 3αµτ h⁻¹)kcuⁿ_hkkuⁿ⁺¹_h −ucⁿ_hk.

Here we use the fact that|s₂| ≤C_wxτ. Then divide both sides of the above inequality bykuⁿ⁺¹_h −ucⁿ_hkto get, kuⁿ⁺¹_h −ucⁿ_hk ≤(Cwxτ+ 3αµτ h⁻¹)kucⁿ_hk, (4.11) which yields that

1

2kuⁿ⁺¹_h −cuⁿ_hk²≤(C_wx² τ²+ 9α²µ²τ²h⁻²)kucⁿ_hk². Under the CFL condition (4.2), we obtain the following inequality,

1

2kuⁿ⁺¹_h −ucⁿ_hk²≤(C_wx² τ²+τ)kcuⁿ_hk²

≤Cτkuⁿ_hk².

Here the last step uses (2.21) andτ≤1. Consequently, the energy identity (4.6) implies that 1

2kuⁿ⁺¹_h k²−1

2kuⁿ_hk²≤Cτkuⁿ_hk². P⁰case. Apply the equivalent form (3.13) of the operatorAto rewrite (4.9),

(uⁿ⁺¹_h −ucⁿ_h,vc_hⁿ)_Kn+1

j =−τB(cuⁿ_h,cvⁿ_h)_Kn+1 j ,

due to the fact that s₂ =τ ω_x(t_n+1). Since the finite element space is piecewise constant, we have ∂_xvⁿ_h = 0, which is not available for k ≥ 1. Take the test function vcⁿ_h = uⁿ⁺¹_h −cuⁿ_h in the above equality and use the boundedness (3.14) of the operatorBto yield,

kuⁿ⁺¹_h −ucⁿ_hk ≤√

2αµτ h⁻¹²[[uⁿ_h]], (4.12)

which leads to

1

2kuⁿ⁺¹_h −ucⁿ_hk²≤α²µ²τ h⁻¹[[uⁿ_h]]². (4.13) If

α²µ²τ²h⁻¹−τ

4α≤0, that is, αµ²τ h⁻¹≤ 1 4, we finish the proof by combining (4.6) and (4.13) together,

1

2kuⁿ⁺¹_h k²−1

2kuⁿ_hk²≤0.

In the following, we provide a remark to summarize the main difference between the caseP⁰ andP^k,k≥1 in the stability analysis of the first order scheme.

Remark 4.2. For the piecewise constant finite element space,V_h(t) ={v(χj(·, t))∈P⁰([−1,1])}, we have the property ∂xvh = 0 with vh ∈ Vh(t), which can not be extended to the finite element space with polynomial degreek≥1. Thus, the bound (4.12) is no longer available for the caseP^k, k≥1. Instead, we use the inverse inequality to control∂_xv_h and get the bound (4.11). In the end, two different CFL conditions are obtained for the stability of the first order scheme.

4.2. Second order scheme

The ALE-DG with TVD-RK2 scheme is given in the following form: finduⁿ⁺¹_h ∈V_h(t_n+1), such that for any v_hⁿ≡v_h(·, tn)∈V_h(t_n) and 1≤j≤N, there hold

(u¹_h,cv_hⁿ)_Kn+1

j = (uⁿ_h, v_hⁿ)_Kⁿ

j −τA(uⁿ_h, vⁿ_h)_Kⁿ

j, (uⁿ⁺¹_h ,cv_hⁿ)_Kn+1

j

= 1

2(uⁿ_h, v_hⁿ)_Kⁿ

j +1

2(u¹_h,cv_hⁿ)_Kn+1

j −τ

2A(u¹_h,cv_hⁿ)_Kn+1 j

.

(4.14)

Herevcⁿ_h is defined by (2.12).

Theorem 4.3. Let uⁿ⁺¹_h be the numerical solution of the fully discrete scheme (4.14), then for any n, there holds

kuⁿ⁺¹_h k²≤(1 +Cτ)kuⁿ_hk², under the CFL condition

τ h^−4/3≤ ³ s 4

81(αµ)⁴·

In particular, for the piecewise linear finite element space, Vh(t) ={v(χj(·, t))∈P¹([−1,1])}, we just need the usual CFL condition,

αµτ h⁻¹≤min 1

32µ, 1

√3

16µ

. (4.15)

Here αis defined by (3.3),µis the inverse constant (2.26), andC is a positive constant depending solely on C_wx andµ.

Proof. Rewrite the scheme (4.14) such that all of the terms are in the same cellK_jⁿ⁺¹, (u¹_h,vcⁿ_h)_Kn+1

j

= (1−s₂)(cuⁿ_h,cv_hⁿ)_Kn+1

j −τA(cuⁿ_h,cv_hⁿ)_Kn+1 j

, (uⁿ⁺¹_h ,vcⁿ_h)_Kn+1

j =1−s2

2 (cuⁿ_h,cv_hⁿ)_Kn+1

j +1

2(u¹_h,vc_hⁿ)_Kn+1

j −τ

2A(u¹_h,vcⁿ_h)_Kn+1 j .

(4.16)

Here we use (4.7) and (4.8). By takingvc_hⁿ = ¹₂ucⁿ_h,u¹_h in the above equalities, respectively, and adding them together, we have

1

2kuⁿ⁺¹_h k²_Kn+1 j

−1−s2

2 kucⁿ_hk²_Kn+1 j

=1

2kuⁿ⁺¹_h −u¹_hk²_Kn+1 j

+s2

4ku¹_h−ucⁿ_hk²_Kn+1 j

−s2

4 kcuⁿ_hk²_Kn+1 j

−s2

4ku¹_hk²_Kn+1 j

−τ

2A(cuⁿ_h,ucⁿ_h)_Kn+1 j

−τ

2A(u¹_h, u¹_h)_Kn+1 j

. (4.17)

Noticing that

kuⁿ_hk²_Kn

j = (1−s2)kucⁿ_hk²_Kn+1 j

, s2=τ ωx(tn+1), (4.18)

we obtain the energy identity by summing (4.17) over allj and using the property (3.18) ofA, 1

2kuⁿ⁺¹_h k²−1

2kuⁿ_hk²= 1

2kuⁿ⁺¹_h −u¹_hk²+

N

X

j=1

s2

4ku¹_h−ucⁿ_hk²_Kn+1 j

−τ

4α[[uⁿ_h]]²−τ

4α[[u¹_h]]². (4.19)

In order to obtain the stability, we just need to analyze the first two terms of the right hand side in the above equality. From (4.16), it is straightforward to get

(u¹_h−ucⁿ_h,cv_hⁿ)_Kn+1

j =−τA(cuⁿ_h,cv_hⁿ)_Kn+1

j −s2(cuⁿ_h,vcⁿ_h)_Kn+1

j , (4.20)

(uⁿ⁺¹_h −u¹_h,cv_hⁿ)_Kn+1

j =−τ

2A(u¹_h−ucⁿ_h,vc_hⁿ)_Kn+1

j . (4.21)

P^k case. Take the test functioncv_hⁿ=uⁿ⁺¹_h −u¹_h in (4.21) and sum up all overj to yield, kuⁿ⁺¹_h −u¹_hk²=−τ

2A(u¹_h−ucⁿ_h, uⁿ⁺¹_h −u¹_h)(tn+1). (4.22) Using the boundedness (3.6) ofA, we have

kuⁿ⁺¹_h −u¹_hk ≤ 3

2αµτ h⁻¹ku¹_h−ucⁿ_hk. (4.23) Then by the similar arguments, taking the test function cvⁿ_h =u¹_h−ucⁿ_h in (4.20) and using the boundedness (3.6) lead to

ku¹_h−ucⁿ_hk ≤(C_wxτ+ 3αµτ h⁻¹)kcuⁿ_hk. (4.24) Denoteλ=αµτ h⁻¹. Combine (4.23) and (4.24) to get that

1

2kuⁿ⁺¹_h −u¹_hk²≤ 9

4λ²(C_wx² τ²+ 9λ²)kucⁿ_hk².