doi: 10.4208/cicp.OA-2017-0204 February 2019
Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type
Equations
Qian Zhang1and Yinhua Xia1,∗
1School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.
Received 24 September 2017; Accepted (in revised version) 12 February 2018
Abstract. In this paper, we develop the Hamiltonian conservative andL2conservative local discontinuous Galerkin (LDG) schemes for the Korteweg-de Vries (KdV) type equations with the minimal stencil. For the time discretization, we adopt the semi- implicit spectral deferred correction (SDC) method to achieve the high order accuracy and efficiency. Also we compare the schemes with the dissipative LDG scheme. Stabil- ity of the fully discrete schemes is provided by Fourier analysis for the linearized KdV equation. Numerical examples are shown to illustrate the capability of these schemes.
Compared with the dissipative LDG scheme, the numerical simulations also indicate that the conservative LDG scheme with high order time discretization can reduce the long time phase error validly.
AMS subject classifications: 65M60, 65M12, 35Q53
Key words: Local discontinuous Galerkin method, conservative and dissipative schemes, Korteweg-de Vries type equations, semi-implicit spectral deferred correction method.
1 Introduction
In this paper, we consider the initial value problem of Korteweg-de Vries (KdV) equation (ut+f(u)x+εuxxx=0, x∈I= [a,b], t>0,
u(x,0) =u0(x), (1.1)
wheret is time,xis the space coordinate in the direction of propagation,a,b,ε∈R,ε>0.
With smooth enough initial conditionu0(x), we can obtain the existence and uniqueness of solution [8]. The KdV equation is first introduced by Boussinesq (1877) and redis- covered by Diederik Korteweg and Gustav de Vries in 1895 [14], in which studies the
∗Corresponding author.Email addresses:[email protected](Q. Zhang),[email protected](Y. Xia)
http://www.global-sci.com/ 532 2019 Global-Science Pressc
small-amplitude long waves in shallow water. In the study of water wave, it has two well-known solutions, cnoidal wave and solitary wave solutions. In the last decades, since the soliton solution proposed by Zabusky and Kruskal [33], the KdV equation has risen a considerable interest by physicists and mathematicians. Actually, the KdV equa- tion is a mathematical model for the propagation of nonlinear dispersive long waves in many branches of physics and engineering including aerology, oceanography, plasma physic, geology, among many others.
Various numerical methods of solving this equation have been proposed, like finite- difference schemes [12, 20], pseudospectral methods [10], heat balance integral method [15] and finite element method, especially discontinuous Galerkin method. The discon- tinuous Galerkin method (DG method) is a class of finite element methods using com- pletely discontinuous piecewise polynomial functions as numerical approximation and test functions. The DG method was first introduced in 1973 by Reed and Hill in [19] for solving steady state linear hyperbolic equations. The important ingredient of this method is the design of suitable inter-element boundary treatments (so called numerical fluxes) to obtain highly accurate and stable schemes in many situations.
Within the DG framework, the method was extended to deal with derivatives of or- der higher than one, i.e. local discontinuous Galerkin (LDG) method. The first LDG method was introduced by Cockburn and Shu in [5] for solving convection-diffusion equation. Their work was motivated by the successful numerical experiments of Bassi and Rebay [3] for compressible Navier-Stokes equations. Later, Yan and Shu developed a LDG numerical method for a general KdV type equation containing third order deriva- tives in [31], and they generalized the LDG method to PDEs with fourth and fifth spatial derivatives in [32]. Levy, Shu and Yan [17] developed LDG methods for nonlinear dis- persive equations that have compactly supported traveling wave solutions, the so-called compactons. More recently, Xu and Shu further generalized the LDG method to solve a series of nonlinear wave equations [24–27]. We refer to the review paper [29] of LDG methods for high-order time-dependent partial differential equations.
According to the selection of numerical flux function for the nonlinear term f(u)and the dispersive termεuxxx in KdV equations, the DG method can be divided into dissi- pative and conservative schemes. Conservative discretization scheme means that this scheme can preserve certain conserved quantities discretely. In the numerical experi- ments of [4], the higher accuracy and better stability of the conservative scheme over long temporal intervals can be seen. Usually, the conservation ofL2energy
H1=
Z 1
2u2dx, (1.2)
and the conservation of the Hamiltonian H2=
Z ε
2u2x−V(u)dx, V(u) =
Z u
f(ζ)dζ, (1.3)
are considered, since the KdV equation is a Hamiltonian system [11].
In [31], Yan and Shu proposed a dissipative LDG method for the KdV equation and proved the L2 sub-optimal error estimates of the semi-discrete LDG numerical method for linear KdV equation. Then Xu and Shu proved the L2 optimal semi-discrete error estimates for the linear KdV equation in [30]. Cheng and Shu developed a new discon- tinuous Galerkin method in [7] to solve KdV equation without the auxiliary variables in LDG numerical method. Based on the results of Cheng and Shu, Bona et al. proposed a global projection in [4] to develop conservative method for generalized KdV type equa- tions. However, this method does not achieve the optimal (k+1)-th order of accuracy when piecewise polynomials of odd degreek is used (this accuracy degeneracy is well known for evenkand is also shown to exist for oddkrecently in [13]). In [16], the authors developed theL2 conservative LDG numerical scheme and compared with dissipative LDG scheme for KdV type equations to display the phase error caused by dissipation.
Compared with the dissipative scheme, theL2conservative in [16] and Hamiltonian con- servative LDG numerical scheme proposed in [18] can both reduce the phase error effi- ciently. For evenk, these L2and Hamiltonian conservative schemes can achieve optimal order of convergence rate, however, for oddk, only reach sub-optimal order. Meanwhile, the conservative LDG numerical schemes in [22, 23] for long wave and short wave inter- action systems appear to have better convergence rate, due to the alternative numerical fluxes are adopted as much as possible. It results in the LDG scheme with the minimal stencil. In this paper we will develop the Hamiltonian conservative andL2conservative LDG schemes for KdV type equations with the minimal stencil. It likely has better ac- curacy for Hamiltonian conservation LDG scheme with the minimal stencil through the numerical experiments in Section 4.
To achieve the high order accuracy and efficiency for the fully discrete scheme, we adopt the spectral deferred correction (SDC) scheme [9]. It is based on low order time integration methods which are corrected iteratively, with the order of accuracy increased for each additional iteration. Due to the third derivative term, explicit time discretiza- tion will suffer from a strict time step restriction for stability. Implicit time discretization like mid-point time discretization scheme can break the time step restriction. Combined with conservative spatial discretization scheme, it can achieve a conservative fully dis- crete scheme which can reduce phase error. However, for nonlinear term f(u), implicit scheme associated with a nonlinear algebraic system will cause computation complica- tion. Thus, semi-implicit SDC scheme is used to solve the KdV type equation in [21] to achieve arbitrary high order accuracy and efficiency. For nonlinear term, we use explicit time discretization while using implicit time discretization for the third order derivative term. When high order semi-implicit SDC schemes combine with conservative LDG nu- merical schemes, it can reduce the phase error efficiently even though the fully discrete scheme is no longer conservative.
The paper is organized as follows. In Section 2, we introduce the LDG numerical method briefly. Combining with different numerical flux functions, we presentL2 con- servative LDG numerical scheme and Hamiltonian conservative LDG numerical scheme respectively. Section 3 is devoted to time discretization methods, especially the semi-
implicit SDC scheme and the Fourier stability analysis of the fully discrete schemes for the linearized KdV equation. And in Section 4 we provide a few numerical experiments which include linear and nonlinear KdV equations to illustrate the accuracy and the long time behavior of the dissipative and conservative LDG schemes. Concluding remarks are given in Section 5.
2 Conservative local discontinuous Galerkin method
In this section, we present and analyze theL2and Hamiltonian conservative local discon- tinuous Galerkin schemes for the KdV equation (1.1) with the period L. Notice that the assumption of periodic boundary conditions is for simplicity only and is not essential.
The method can be easily designed for non-periodic boundary conditions.
We denote the meshThby Ij= [xj−1
2,xj+1
2]forj=1,···,N, with the cell center denoted byxj=12(xj−1
2+xj+1
2). The cell size is∆xj=xj+1
2−xj−1
2 andh=max1≤j≤N ∆xj. We define the finite element space as the solution and test function space consisting of piecewise polynomials
Vhk={v:v|Ij∈Pk(Ij); 1≤j≤N},
wherePk(Ij)denotes the set of polynomial of degree up tokdefined on the cell Ij. Note that functions in Vhk are allowed to have discontinuous across cell interfaces. We also denote by the u−j+1
2 andu+j+1
2 the values of u at xj+1
2, from the left cell Ij and the right cellIj+1respectively. And we define the jump ofuas[[u]] =u+−u−, the average ofuas {{u}}=12(u++u−). For simplicity, we just setε=1 in the description of Section 2 and Section 3.
Following the framework of LDG numerical method [29], we first rewrite the KdV equation (1.1) as a first-order system:
ut+f(u)x+wx=0, w−vx=0,
v−ux=0.
(2.1)
Then the LDG scheme to the solve the first-order KdV system (2.1) is as follows: finduh, whandvh∈Vhksuch that
((uh)t,φ)Ij+<\f(uh),φ>I
j−(f(uh),φx)Ij+<wch,φ>I
j−(wh,φx)Ij=0,
(wh,ϕ)Ij−<vbh,ϕ>I
j+(vh,ϕx)Ij=0,
(vh,ψ)Ij−<ubh,ψ>I
j+(uh,ψx)Ij=0,
(2.2)
for all test functionsφ,ϕ,ψ∈Vhk, andIj∈ Th. Here, we adopt the round bracket and angle
bracket to simplify the expressions:
(u,v)Ij=
Z
Ijuvdx,
<u,ˆ v>I
j=uˆj+1
2v−j+1 2
−uˆj−1
2v+j−1 2,
(2.3)
for one dimension case. The “hat” terms in (2.2) are the so-called numerical fluxes which are functions defined on the cell boundary from integration by parts and should be de- signed based on different guiding principles for different PDEs to ensure stability and local solvability of the intermediate variables.
We introduce the following operators to simplify our notations:
Definition 2.1. We define the operatorsL±j ,c(·,·)as follows:
L+j (u,v) =−(u,vx)Ij+<u+,v>I
j, (2.4)
L−j (u,v) =−(u,vx)Ij+<u−,v>I
j, (2.5)
Lcj(u,v) =−(u,vx)Ij+<{{u}},v>I
j (2.6)
for∀u,v∈Vhk.
Definition 2.2. The operatorsNjc,d(·,·)for the nonlinear term f(u)of KdV equation are defined:
Ndj(u,φ) =−(f(u),φx)Ij+<[f(u),φ>I
j, (2.7)
where[f(u) =12(f(u+)+f(u−)−α(u+−u−)),α=maxu|f′(u)|, it is dissipative treatment for the nonlinear term. Then we define the conservative treatment for f(u)
Njc(u,φ) =−(f(u),φx)Ij+<[f(u),φ>I
j, (2.8)
where[f(u)is taken as conservative flux function [f(u) =
([[F(u)]]
[[u]] , [[u]]6=0,
f({{u}}), [[u]] =0., (2.9) whereF(u) =Ru
f(τ)dτ, especially for f(u) =up, pis integer, [f(u) = 1
p+1
p
∑
j=0
(u+)p−j(u−)j, for∀u,v∈Vhkas in [4].
Lemma 2.1. Let L±,c=∑jL±j ,c, Nc,d=∑jNjc,d, there holds the equalities
Lc(u,u) =0, (2.10)
L−(u,u) =1
2[[u]]2, (2.11)
L+(u,v) =−L−(v,u), (2.12) Lc(u,v) =−Lc(v,u), (2.13)
Nd(u,u)≥0, (2.14)
Nc(u,u) =0, (2.15)
for∀u,v∈Vhk.
These properties in Lemma (2.1) are listed in [28], [4], so we do not give the detail here.
Then the dissipative LDG numerical scheme in [31] could be written as
((uh)t,φ)Ij+Njd(uh,φ)+L+j (wh,φ) =0, (wh,ϕ)Ij−L+j (vh,ϕ) =0,
(vh,ψ)Ij−L−j (uh,ψ) =0,
(2.16)
which can also be written as the scheme (2.2) with the numerical fluxes c
wh=w+h, vbh=v+h, ubh=u−h (2.17) and [f(u) in (2.7). Here, we remark that the numerical fluxes we take above are not unique. The key for third derivative term is the opposite sides betweenubh andwch. vbh
depends on the sign ofε. We assume thatε>0 in this paper, then for dissipative numer- ical fluxes, vbh should be taken as v+h. For the nonlinear term f(u), flux function for the dissipative LDG numerical scheme can also be Godunov, Boltzmann and Engquist-Osher flux function.
In [31], the dissipative LDG numerical fluxes have been adopted for KdV equation.
For hyperbolic conservation laws, the dissipative numerical fluxes are chosen such that the entropy condition can be satisfied. As an additional condition, entropy condition leads us to single out entropy solution among the infinitely weak solutions. Analogously, the dissipation introduced by the LDG numerical scheme for the KdV type equation is es- sential to ensure stability for the odd derivatives which correspond to waves. However, the KdV type equation profiles the propagation of nonlinear, dispersive waves which cause the energy conservation of general initial data. The dissipative LDG numerical scheme will destroy the balance between nonlinear steepening and dispersive spreading.
Numerically, it will cause the phase error, shape error and the inaccuracy of the LDG numerical scheme over a long temporal interval. In [4], [16], [18], the authors developed
some different kinds of conservative schemes (details will be discussed presently) for the KdV type equation. The conservative scheme means that it can preserve the conserved quantities discretely so that the numerical schemes can approximate the conservative form of the KdV equation. Comparing with the dissipative scheme, the conservative dis- cretization scheme not only has high accuracy and good stability, but reduces the phase error and shape error validly over a long temporal interval, especially in low order ap- proximation.
Proposition 2.1. (L2 stability for the dissipative scheme (2.16)) First, we give the cell entropy inequality
d dt
Z
K
u2h(x,t)
2 dx+Fˆj+1
2−Fˆj−1
2≤0, (2.18)
where numerical entropy flux ˆFj+1
2 is Fˆj+1
2=Fˆ
uh(x−j+1
2,t),wh(x−j+1
2,t),vh(x−j+1
2,t), uh(x+j+1
2,t), wh(x+j+1
2,t), vh(x+j+1
2,t). (2.19) Adding up (2.18) over allIj∈ Th, we haveL2stability
∑
j
((uh)t,uh)Ij≤0. (2.20) The entropy inequality and stability of the dissipative LDG numerical scheme for the KdV type equation are mentioned in [31], in which the L2 sub-optimal error estimate of the semi-discrete dissipative LDG numerical scheme for the linear KdV type equation have been proved. In [30], Xu and Shu proved theL2optimal semi-discrete error estimate of the dissipative LDG numerical scheme for the linear KdV type equation.
We denote this dissipative LDG numerical scheme by NC-NC scheme in the section of numerical experiments, which means we treat nonlinear term f(u)and third derivative term both non-conservative.
2.1 L2Conservative LDG numerical scheme
To construct aL2conservative LDG numerical scheme, we choose the following numeri- cal fluxes in (2.2)
[f(u) =
([[F(u)]]
[[u]] , [[u]]6=0,
f({{u}}), [[u]] =0, wch=w−h, vbh={{vh}}, ubh=u+h (2.21) as a replacement, we could also take
[f(u) =
([[F(u)]]
[[u]] , [[u]]6=0,
f({{u}}), [[u]] =0, wch=w+h, vbh={{vh}}, ubh=u−h, (2.22)
where
F(u) =
Z u
f(τ)dτ. (2.23)
The main differences between the numerical fluxes (2.21) and Karakashian and Xing [16], Liu and Yi [18], are the choice ofwch andubh. According to our notations, their nu- merical fluxes ofL2conservative LDG numerical scheme can be presented as
[f(u) =
([[F(u)]]
[[u]] , [[u]]6=0,
f({{u}}), [[u]] =0, wch={{wh}}, vbh={{vh}}, ubh={{uh}}. (2.24) With these numerical fluxes (2.21), Eq. (2.2) can be rewritten as
((uh)t,φ)Ij+Njc(uh,φ)+L+j (wh,φ) =0, (wh,ϕ)Ij−Lcj(vh,ϕ) =0,
(vh,ψ)Ij−L−j (uh,ψ) =0,
(2.25)
for all test functionsφ,ϕ,ψ∈Vhk, and Ij∈ Th.
Proposition 2.2. We get theL2conservative LDG numerical scheme (2.25). The following equality can be established
∑
j
((uh)t,uh)Ij=0, that is,H1in (1.2) is a conservative quantity numerically.
Proof. By choosingφ=uh,ϕ=vhandψ=−whin the scheme (2.25), we have ((uh)t,uh)Ij+Njc(uh,uh)+L+j (wh,uh) =0,
(wh,vh)Ij−Lcj(vh,vh) =0,
−(vh,wh)Ij+L−j (uh,wh) =0.
After summation overIj∈ Th, we can get
∑
j
Njc(uh,uh)+L+j (wh,uh)+L−j (uh,wh)+Lcj(vh,vh) =0.
By Lemma 2.1, we obtain
∑
j
((uh)t,uh)Ij=0.
The proof is completed.
We denote this L2 conservative LDG numerical scheme by C-C scheme. ForL2con- servative LDG numerical scheme with numerical fluxes (2.24),L2optimal error estimates are obtained for the linear KdV equation in [16, 18] whenkis even and Nis odd. More generally, theL2 conservative numerical fluxes can be defined as wch={{wh}}+α[[wh]]
andubh={{uh}}−α[[uh]] in (2.21). Whenα6=±12, it results a wider stencil LDG scheme than the scheme (2.25).
To compare the differences among spatial discretization schemes, we introduce NC-C scheme which treatsf(u)non-conservative and the third derivative termεuxxxconserva- tive. The semi-discrete LDG numerical scheme can be written as
((uh)t,φ)Ij+Njd(uh,φ)+L+j (wh,φ) =0, (wh,ϕ)Ij−Lcj(vh,ϕ) =0,
(vh,ψ)Ij−L−j (uh,ψ) =0,
(2.26)
for all test functionsφ,ϕ,ψ∈Vhk, andIj∈ Th. It is obvious that this NC-C scheme is also dissipative.
2.2 Hamiltonian conservative LDG numerical scheme
In this section, we introduce the Hamiltonian conservative LDG numerical scheme. In [18], Liu and Yi presented a Hamiltonian conservative LDG numerical scheme for the KdV type equation. First we rewrite the KdV equation (1.1) into a first order system
ut+px+wx=0, w−vx=0, v−ux=0, p−f(u) =0.
(2.27)
Then the semi-discrete Hamiltonian conservative LDG numerical scheme is defined: find uh,wh, andvh∈ Vhksuch that
((uh)t,φ)Ij−Lcj(wh,φ) =0,
(wh,ϕ)Ij+Lcj(vh,ϕ)+(f(uh),ϕ)Ij=0, (vh,ψ)Ij−Lcj(uh,ψ) =0,
(2.28)
for all test functionsφ,ϕ,ψ∈Vhk, andIj∈Th. For this LDG numerical scheme, the conserved quantity H2 is conservative numerically. Unlike the above scheme (2.28), we adopt dif- ferent numerical fluxes as follows:
b
ph={{ph}}, wch={{wh}}, vbh=v−h, ubh=u+h. (2.29)
Then the semi-discrete Hamiltonian conservative LDG numerical scheme is defined as:
finduh,wh,vhandph∈ Vhk such that
((uh)t,φ)Ij+Lcj(ph,φ)+Lcj(wh,φ) =0, (wh,ϕ)Ij−L−j (vh,ϕ) =0,
(vh,ψ)Ij−L+j (uh,ψ) =0, (ph,η)Ij−(f(uh),η)Ij=0,
(2.30)
for all test functionsφ,ϕ,ψ,η∈Vhk, andIj∈ Th.
Proposition 2.3. For Hamiltonian conservative LDG numerical scheme (2.30), there holds an equality
∑
j
((vh)t,vh)Ij−(f(uh),(uh)t)Ij=0, (2.31) that is,H2in (1.3) is a conservative quantity numerically.
Proof. Letqh∈Vhk, such that
(Lcj(qh,ξ) = (uh,ξ)Ij, R
Iqhdx=0 (2.32)
for all test functionsξ∈Vh, andIj∈ Th. Taking the time derivative of (2.32) and Choosing ξ=ph,whand(qh)tin (2.32) respectively, we obtain
Lcj((qh)t,ph) = ((uh)t,ph)Ij, Lcj((qh)t,wh) = ((uh)t,wh)Ij, Lcj((qh)t,(qh)t) = ((uh)t,(qh)t))Ij.
(2.33)
Taking the time derivative of the third equation of (2.30) andφ= (qh)t, ϕ= (uh)t,ψ=vh, η= (uh)t, we have
((uh)t,(qh)t)Ij+Lcj(ph,(qh)t)+Lcj(wh,(qh)t) =0, (wh,(uh)t)Ij−L−j (vh,(uh)t) =0,
((vh)t,vh)Ij−L+j ((uh)t,vh) =0, (ph,(uh)t)Ij−(f(uh),(uh)t)Ij=0.
(2.34)
Summing up all equalities in (2.34) overIj∈ Th, we get
∑
j
((uh)t,(qh)t)Ij+Lcj(ph,(qh)t)+((uh)t,ph)Ij
+Lcj(wh,(qh)t)+((uh)t,wh)Ij−(L+j ((uh)t,vh)+L−j (vh,(uh)t))
+(vh,(vh)t)Ij−(f(uh),(uh)t)Ij=0. (2.35)
Finally, according to the equalities (2.33) and Lemma 2.1, we obtain
∑
j
((vh)t,vh)Ij−(f(uh),(uh)t)Ij=0. (2.36) The proof is completed.
We denote the Hamiltonian conservative LDG numerical scheme by HC scheme. For the linear KdV equation, the schemes in [16, 18] are the same. To obtain the Hamiltonian conservation, numerical fluxesvbh andubh in (2.29) can also chosen asvbh={{vh}}+α[[vh]]
andubh={{uh}}−α[[uh]], which results a wider stencil LDG scheme than the scheme (2.30) whenα6=±12.
3 Time discretization
In this section, we mainly utilize semi-implicit SDC schemes. The initial value problem for a first-order system of ODEs takes the form
(ut=FE(t,u(t))+FI(t,u(t)), t∈[0,T],
u(0) =u0, (3.1)
whereFE is a non-stiff term and FI is a stiff term. The subscripts refer that the non-stiff term is treated explicitly and the stiff term is treated implicitly. For the KdV type equation (1.1)
FE=f(u)x; FI=εuxxx; F,FE+FI, (3.2) resulting a linear implicit scheme.
We divide the temporal interval [0,T] into N subintervals, 0=t0<t1<···<tn<···<
tN=T. And we denote∆tn=tn+1−tn. un is the numerical approximation of u(tn)and u0=u(0).
3.1 Semi-implicit SDC scheme
The spectral deferred correction method (SDC) was presented by Dutt, Greengard and Rokhlin in [9]. It is based on low order time integration methods which are corrected iteratively, with the order of accuracy increased for each additional iteration. In [21], Xia et al. developed three different time discretization schemes including semi-implicit SDC scheme for solving the stiff ODEs resulting from a LDG spatial discretization to PDEs containing high order spatial derivatives. Semi-implicit implies that we will split stiff and non-stiff terms as needed, and discretize implicitly for the stiff terms and explicitly for the non-stiff terms. This treatment enlarges time steps∆t=O(∆x) rather than the much more restrictive ∆t=O(∆xk)of explicit time discretization for k-th order PDEs.
Also, the linear implicit scheme reduces the complexity of computation comparing to the nonlinear implicit scheme.
We rewrite the equation (3.1) into an integral form in the subinterval[tn,tn+1]: u(tn+1) =u(tn)+
Z tn+1
tn
F(τ,u(τ))dτ. (3.3)
In[tn,tn+1], choosing the pointstn,mform=0,1,···,Psuch that
tn=tn,0<tn,1<···<tn,m<···<tn,P=tn+1, (3.4)
∆tn,m=tn,m+1−tn,m, ukn,m≃u(tn,m), (3.5) whereukn,m is thek-thorder approximation to u(tn,m). Generally, we choose the Gauss- Lobatto nodes as tn,m in [tn,tn+1]. Gauss-Legendre, Gauss-Radau or Chebyshev nodes can be also used to avoid the instability of approximation at equispaced nodes for high order accuracy.
The algorithm is given as following:
We use the forward/backward Euler method for non-stiff/stiff terms to compute an approximation solutionu1(tn,m),m=0,···,P.
u1n,0=un.
Form=0,···P−1
u1n,m+1=u1n,m+∆tn,mFE(tn,m,u1n,m)+∆tn,mFI(tn,m+1,u1n,m+1).
End for (3.6)
Then we compute the successive corrections Fork=1,···,K
ukn,0+1=un.
Form=0,···P−1
ukn,m+1+1=ukn,m+1+θ1∆tn,m(FE(tn,m,ukn,m+1)−FE(tn,m,ukn,m))
+θ2∆tn,m(FI(tn,m+1,ukn,m+1+1)−FI(tn,m+1,ukn,m+1))+Imm+1(F(t,uk)). End for
End for
un+1=uKn,P+1 (3.7)
where 0≤θ1,θ2≤1. In Section 4, we adoptθ1=0, θ2=1 mostly. There is a few comments on the form of the quadrature forImm+1(F(t,uk)). The pointstn,mlies in the interior of the interval[tn,tn+1], the quadrature must be done for each[tm,tm+1], there arePquadrature rules
Imm+1(F(t,uk)) =
P
∑
l=0
qlmF(tl,ukl), m=1,···,P−1. (3.8)
We can precomputed the coefficients qlm, and the quadrature is reduced to a simple matrix-vector multiplication.
3.2 Fourier analysis of fully discrete schemes
After the introduction of space and time discretization, we attempt to give an stability analysis of fully discrete scheme for linear KdV equation
ut+ux+uxxx=0. (3.9)
Fourier expansion method in [34] can be used to analyze the stability. Above four schemes in Section 2 are semi-discrete schemes, i.e. the conservativeness is a spatial property. Time discretization need to be added to analyze the fully discrete scheme and fully conserva- tiveness. But as a result of non-conservativeness of semi-implicit SDC time discretization method, generally, the fully discrete schemes are not completely conservative. So these four schemes have different degrees of dissipation or ascent which can have some ef- fect on the numerical results. To compare, we also list a conservative time discretization scheme called mid-point scheme [4]. Combining with C-C spatial discretization scheme, we get aL2conservative fully discrete scheme. In Section 4, we also list a few results that use IMEX Runge-Kutta schemes [2] as time discretization scheme.
First, we need rewrite the formulation of the LDG numerical scheme as finite differ- ence scheme, and then we use finite difference techniques to analyze stability. Towards this goal we choose the degrees of freedom for thek-th degree polynomial inside the cell Ij as the point values of solution, denoted by
uj+ 2i−k
2(k+1), i=0,···,k, (3.10) at thek+1 equally spaced nodes
j+ 2i−k 2(k+1)
∆x, i=0,···,k. (3.11)
According to these degrees of freedom, the discontinuous Galerkin method become finite difference scheme on a global uniform mesh with a mesh size∆x/(k+1). We takek=1 as a example, the degrees of freedom are the value of spaced node
uj−1
4,uj+1
4, j=1,···,N. (3.12)
Solution inside the cellIjis represented by u(x) =uj−1
4φj−1
4+uj+1
4φj+1
4, (3.13)
where
φj−1 4(xj−1
4) =1, φj−1 4(xj+1
4) =0, φj+1
4(xj−1
4) =0, φj+1 4(xj+1
4) =1. (3.14)
We take the test functions also as φj−1
4(x)and φj+1
4(x), then we obtain easily the finite difference scheme.
We rewrite the NC-NC scheme(2.16) in matrix form M vj−1
4
vj+1
4
!
= 1
∆x
"
A1
uj−1
4
uj+1
4
! +B1
uj−5
4
uj−3
4
!#
, M wj−1
4
wj+1
4
!
= 1
∆x
"
A2
vj−1
4
vj+1
4
! +B2
vj+3
4
vj+5
4
!#
, M u′j−1
4
u′j+1 4
!
=− 1
∆x
"
A1
uj−1
4
uj+1
4
! +B1
uj−5
4
uj−3
4
!#
− 1
∆x
"
A2
wj−1
4
wj+1
4
! +B2
wj+3
4
wj+5
4
!#
(3.15)
for j=1,···,N. Here we denote the time derivative ofuby u′, M denotes mass matrix.
And A1,B1 denote the coefficient matrixes of numerical flux L−j and the integral term, respectively;A2,B2denote those of numerical fluxL+j and the integral term, respectively.
A1=
5
4 1
4
−74 54
!
, B1=
3 4 −94
−14 34
! , A2= −54 74
−14 −54
!
, B2= −34 14
9 4 −34
!
, M=∆x
7
12 −121
−121 127
! . We can write (3.15) into a more compact form
u′j−1 4
u′j+1 4
!
=− ∆x1 M−1
"
A1
uj−1
4
uj+1
4
! +B1
uj−5
4
uj−3
4
!#
− ∆1x3M−1
"
P1
uj+7
4
uj+9
4
! +P2
uj+3
4
uj+5
4
! +P3
uj−1
4
uj+1
4
! +P4
uj−5
4
uj−3
4
!#
, (3.16) whereP1,P2,P3,P4are the product ofA1,A2,B1,B2. For (3.16), Fourier analysis can be used to prove stability. We make an ansatz of the form
unj−1 4
(t) unj+1
4
(t)
!
= uˆ
n m,−41(t) ˆ unm,1
4
(t)
!
eimxj, (3.17)
and substitute this form into the (3.16) and we can get ˆ
u′m,−1 4
(t) ˆ u′m,1
4
(t)
!
=G(m,∆x) uˆm,−41(t) ˆ um,1
4(t)
!
, (3.18)
where G(m,∆x) is called the amplification factor matrix, m is a integer which denotes frequency (or wavenumber) of wave. And then we consider the fully discrete scheme, for
