
We study Hom-actions, semidirect product and describe the relation between semi-direct product extensions and split extensions of Hom-preLie algebras. We obtain the functorial properties of the universal α-central extensions of α-perfect Hom-preLie algebras. We give that a derivation or an automorphism can be lifted in an α-cover with certain constraints. We provide some necessary and sufficient conditions about the universal α-central extension of the semi-direct product of two α-perfect Hom-preLie algebras.
Citation: Bing Sun, Liangyun Chen, Yan Cao. On the universal α-central extensions of the semi-direct product of Hom-preLie algebras[J]. Electronic Research Archive, 2021, 29(4): 2619-2636. doi: 10.3934/era.2021004
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We study Hom-actions, semidirect product and describe the relation between semi-direct product extensions and split extensions of Hom-preLie algebras. We obtain the functorial properties of the universal α-central extensions of α-perfect Hom-preLie algebras. We give that a derivation or an automorphism can be lifted in an α-cover with certain constraints. We provide some necessary and sufficient conditions about the universal α-central extension of the semi-direct product of two α-perfect Hom-preLie algebras.
In this paper, we are interested in semi-Lagrangian (SL) finite difference (FD) schemes for solving nonlinear hyperbolic conservation laws (HCL)
{ut+f(u)x=0,u(x,0)=u0(x). | (1) |
The SL scheme updates the solution by following the characteristics, either forwardly [10,25], or backwardly [19,14], where for (1) the characteristic equation is defined as
dxdt=f′(u(x,t)). | (2) |
The classical backward SL scheme traces the characteristics back to a previous time level and then updates the solution with interpolation, or polynomial reconstruction. Based on its solution space, there are SL FD schemes [26,29], SL finite volume (FV) schemes [9,12], SL spectral element methods [32,13], SL discontinuous Galerkin (DG) finite element methods [23,3], SL particle methods [8], etc.
For SL schemes, mass conservation is an important property for certain applications, such as in kinetic simulations [9,2], weather forecasting [27,15], fluid dynamics [29,12], etc. For finite volume schemes and finite element methods, it is more natural to enforce the discrete mass conservation, by working with the integral form for finite volume schemes [12], or the weak form for finite element methods [2]. For finite difference schemes, it is much more challenging [18]. A conservative SL finite difference scheme based on the flux difference form was proposed in [21]. A maximum principle preserving limiter was applied to ensure the
For SLFD schemes solving nonlinear hyperbolic problems, the main difficulty lies on the characteristic speed depending on the unknown solution, such as in (2), especially for high dimensional problems. Except the predictor-corrector approach in [20], there are also some other methods proposed to achieve high order in time. One is the class of multi-step schemes based on backward differential formulas (BDF), such as the Adams-Moulton and Adams-Bashforth schemes [14]. Another is based on commutator-free Runge-Kutta exponential integrators (RKEI) [5,6]. However, both methods in literature do not have mass conservation. In this paper, we will adopt the RKEI method and combine it with the SLFD scheme in [21] to develop conservative SLFD-RKEI schemes for solving the nonlinear hyperbolic problems. The RKEI method has also recently been coupled with SLDG scheme for solving the Vlasov equations [2].
The RKEI method was first proposed in [7], which is designed for nonlinear ordinary differential equations (ODEs). The idea is to mimic the exponential solution for linear ODEs, and write the nonlinear function as a coefficient multiplying the unknown solution. Then the coefficient is frozen at previous given time levels, so that the resulting ODE have explicit exponential solutions. Later, the RKEI method was realized to be equivalent to the semi-Lagrangian scheme for partial differential equations (PDEs) [5,6]. In this paper, we propose to combine it with the conservative SLFD scheme [21] under the flux difference form. However, different from [31], the temporal integral is transferred to a spatial integral at the previous time level with know solutions, owing to the divergence theorem. WENO interpolations and numerical flux reconstructions [21] are used to ensure essentially non-oscillatory properties. Other methods, such as Hermite WENO schemes (HWENO) [4] may also be applied. However, we would emphasize that in order to achieve large time step condition, we have to combine the WENO interpolation and flux reconstruction, so that the common integration in the left and right fluxes can be canceled exactly [21,18]. This cancellation typically due to the fact, for the one-dimensional problem the spatial integrals in the reconstructions of the left and right fluxes are along the same line (only one dimension). This is not the case for two and higher dimensions. The extension to high dimensional problems is not straightforward. Although dimensional splittings can still be used, they suffer from low order splitting errors and we do not pursue them here.
The rest of the paper is organized as follows. In Section 2, the model equation will be described. The SLFD scheme in [21] and the RKEI method [7] will first be recalled in Section 3, after that conservative SLFD schemes for solving (1) will be presented. In Section 4, numerical experiments will be performed to demonstrate the effectiveness of our proposed methods. Conclusion remarks will be made in Section 5.
We start with the hyperbolic equations in the advective form,
ut+V(u)(x,t)ux=0, | (3) |
where
ddtx(t)=V(u)(x(t),t). | (4) |
When
However, numerical schemes designed according to the advective form (3) cannot provide a correct shock speed for shock waves. It is important instead using the following conservative form
ut+(a(u)(x,t)u)x=0. | (5) |
(5) shares the same characteristic equation as (3) for the same problem. But only in the linear or quasilinear case,
ut+(u22)x=0, | (6) |
In the following, for our numerical scheme design, we consider (4) as the characteristic equation for (3). For the conservative equation (5), instead of (4), we use a pseudo characteristic equation
ddtx(t)=a(u)(x(t),t). | (7) |
Taking the Burgers' equation as an example, we consider
ddtx(t)=u(x(t),t), | (8) |
for (3), while
ddtx(t)=12u(x(t),t), | (9) |
for (5). Similarly for other nonlinear hyperbolic equations.
In this section, we will describe how to design conservative SLFD WENO schemes, based on the RKEI method. We first recall the conservative SLFD scheme defined for the quasilinear hyperbolic equation in [21], then we will present the framework of RKEI method proposed in [5]. After that, we will show how to get conservative schemes by coupling these two methods.
The conservative SLFD scheme starts from the conservative equation (5). Assuming we already have the solution at time level
u(x,tn+1)=u(x,tn)−(∫tn+1tna(u)(x,t)udt)x. | (10) |
Denoting
H(x)=∫tn+1tna(u)(x,t)udt, | (11) |
and defining
H(x)=1Δx∫x+Δx2x−Δx2h(ξ)dξ, | (12) |
we obtain
Hx=1Δx(h(x+Δx2)−h(x−Δx2)), | (13) |
where
At a given point
u(xi,tn+1)=u(xi,tn)−1Δx(h(xi+Δx2)−h(xi−Δx2)). | (14) |
A conservative finite difference scheme can be provided as follows
un+1i=uni−1Δx(ˆhi+12−ˆhi−12), | (15) |
where
Hi=H(xi)=∫tn+1tna(u)(xi,t)u(xi,t)dt. | (16) |
It is important on how to get high order approximations of (16), as we only know the solution at time level
Let's introduce a spatio-temporal region
Now if we integrate the conservative equation (5) over the region
∫Ωi,tut+(a(u)(x,t)u)x=0. | (17) |
Denoting
0=∫Ωi,tut+(a(u)(x,t)u)x=∫Ωi,t∇t,x⋅(u,a(u)(x,t)u)=∫∂Ωi,t(u,a(u)(x,t)u)⋅n=−∫xix∗iu(x,tn)dx+∫tn+1tna(u)(xi,t)u(xi,t)dt, |
where
Hi=∫tn+1tna(u)(xi,t)u(xi,t)dt=∫xix∗iu(x,tn)dx. | (18) |
So if we can solve the characteristic equation (7) to find the root
un+1=SLFDc(a(u)(x,t),Δt)un. | (19) |
In [21], only
On the other hand, another type of SLFD scheme can base on the nonconservative advective form (3). This scheme is to solve the characteristic equation (4) to get
un+1=SLFDn(V(u)(x,t),Δt)un. | (20) |
The nonconservative approach cannot get a right shock speed for nonlinear hyperbolic equations. Moreover, it is not appropriate for long time simulations [19]. In our numerical schemes proposed in the following, we could combine "SLFDc" and "SLFDn" to achieve a high order conservative SLFD scheme for the nonlinear hyperbolic equations.
In this subsection, we first briefly review the RKEI method, which was proposed to solve nonlinear ODEs [7]. For more details, we refer to [7,5,2].
We consider a nonlinear initial value ODE problem of size
dY(t)dt=C(Y)Y,Y(t=0)=Y0, | (21) |
where
dY(t)dt=CnY,Yn+1=exp(CnΔt)Yn, | (22) |
where
To achieve high order accuracy, a class of commutator-free RKEI methods with multi-stages can be used. For an
Algorithm 1: The commutator-free (CF) RKEI method [7]. |
for end |
In the subindex of
Example 3.1. A 3rd order RKEI method in [5] can be represented by the following Butcher tableau,
with which, the RKEI scheme for the nonlinear ODE system (21) reads
Y(1)=YnY(2)=exp(12ΔtC(Y(1)))YnY(3)=exp(Δt(−C(Y(1))+2C(Y(2))))YnYn+1=exp(Δt(112C(Y(1))+13C(Y(2))+512C(Y(3))))exp(Δt(112C(Y(1))+13C(Y(2))−14C(Y(3))))Yn, |
here
Example 3.2. A 4th order RKEI method in [5] can be represented by the following Butcher tableau,
correspondingly, the RKEI scheme for the nonlinear ODE system (21) reads
Y(1)=YnY(2)=exp(12ΔtC(Y(1)))YnY(3)=exp(12ΔtC(Y(2)))YnY(4)=exp(Δt(−12C(Y(1))+C(Y(3))))exp(12ΔtC(Y(1)))YnYn+1=exp(Δt(−112C(Y(1))+16C(Y(2))+16C(Y(3))+14C(Y(4))))exp(Δt(14C(Y(1))+16C(Y(2))+16C(Y(3))−112C(Y(4))))Yn, |
here
The RKEI method for linear ODEs is later recognized to be equivalent to an SL scheme for updating the solution for linear transport problems. The method is then coupled with spectral Galerkin method and applied to convection-diffusion problems [7], incompressible Navier-Stokes equations [6]. Very recently, it has been coupled with SLDG methods, and applied to the Vlasov-Poisson system and guiding-center model problems [2].
Here we will couple the RKEI method with the SLFD scheme, to get a high order SLFD-RKEI scheme for solving the nonlinear hyperbolic equations. The difficulty here is that for SL schemes under the finite difference framework, it is not easy to achieve the following properties simultaneously, high order, conservative, large time steps and ability to solve nonlinear hyperbolic equations. Our method proposed here for the one-dimensional hyperbolic equations, will have these properties.
Two types of conservative schemes can be proposed. One is that we only do a conservative correction at the last stage as compared to a nonconservative one. The other is we do it at every stage. Taking the 3rd order RKEI scheme in Table 2 as an example, the first SLFD-RKEI scheme reads as follows
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u(1)=unu(2)=SLFDn(12V(u(1))(x,tn),Δt)unu(3)=SLFDn(−V(u(1))(x,tn)+2V(u(2))(x,tn+12Δt),Δt)unun+1=SLFDc(112a(u(1))(x,tn)+13a(u(2))(x,tn+12Δt)+512a(u(3))(x,tn+Δt),Δt)SLFDc(112a(u(1))(x,tn)+13a(u(2))(x,tn+12Δt)−14a(u(3))(x,tn+Δt),Δt)un. | (23) |
We denote this scheme as "SLFDc1". The other is
u(1)=unu(2)=SLFDc(12a(u(1))(x,tn),Δt)un |
u(3)=SLFDc(−a(u(1))(x,tn)+2a(u(2))(x,tn+12Δt),Δt)unun+1=SLFDc(112a(u(1))(x,tn)+13a(u(2))(x,tn+12Δt)+512a(u(3))(x,tn+Δt),Δt)SLFDc(112a(u(1))(x,tn)+13a(u(2))(x,tn+12Δt)−14a(u(3))(x,tn+Δt),Δt)un. | (24) |
and we denote it as "SLFDc2". The nonconservative scheme with SLFDn at every stage is still denoted as "SLFDn".
We note that for above schemes at each stage, we freeze
● for a first order RKEI scheme, it has only one stage, the characteristic equation is
w(X(t))=a(u)(X(t),tn), | (25) |
{dX(t)dt=w(X(t)),X(tn+1)=xi. | (26) |
It can be solved with an Euler forward scheme, that is, we let
x∗i=xi−w(xi)Δt=xi−a(u)(xi,tn)Δt. | (27) |
After that we call "SLFDc" to update
● for a 3rd order RKEI scheme, it has three stages. For each stage, we have:
- Stage 1: The characteristic equation is
w(X(t))=12V(u)(X(t),tn), | (28) |
{dX(t)dt=w(X(t)),X(tn+1/2)=xi. | (29) |
We solve it with a 3rd order Runge-Kutta scheme to get
{x∗,(1)i=X(tn)u(1)i=Iu(x∗,(1)i,tn)≈u(xi,tn+12). | (30) |
- Stage 2: The characteristic equation is
w(X(t))=−V(u)(X(t),tn)+2V(u)(X(t),tn+12Δt)), | (31) |
{dX(t)dt=w(X(t)),X(tn+1)=xi. | (32) |
We solve it with a 3rd order Runge-Kutta scheme to get
{x∗,(2)i=X(tn)u(2)i=Iu(x∗,(2)i,tn)≈u(xi,tn+1). | (33) |
- Stage 3: This stage has two steps. The first step has the characteristic equation
w(X(t))=112(a(u)(X(t),tn)+4a(u)(X(t),tn+12Δt)−3a(u)(X(t),tn+Δt)), | (34) |
{dX(t)dt=w(X(t)),X(tn+1)=xi. | (35) |
We solve it with a 3rd order Runge-Kutta scheme to get
x∗i=X(tn). | (36) |
Then we call "SLFDc" to obtain
w(X(t))=112(a(u)(X(t),tn)+4a(u)(X(t),tn+12Δt)+5a(u)(X(t),tn+Δt)), | (37) |
{dX(t)dt=w(X(t)),X(tn+1)=xi. | (38) |
Similarly, we solve it with a 3rd order Runge-Kutta scheme to get
x∗i=X(tn), | (39) |
and we call "SLFDc" to update
In Stage 1 and Stage 2,
A similar procedure can be applied to other types of RKEI schemes, e.g., the 4th order in Table 3. "SLFDc2" will have every stage as Stage 3 above. Here we all omit them to save space.
- |
||||
In this section, numerical examples for one dimensional linear and nonlinear hyperbolic equations are presented to verify the effectiveness of our algorithms. Three schemes are considered. For the 3rd order RKEI scheme (2), "SLFDc1" corresponds to (23), "SLFDc2" corresponds to (24), and "SLFDn" will have "SLFDn" instead of "SLFDc" in the last stage of (23). The schemes with 4th order RKEI (3) can be defined correspondingly.
The formal 5th order finite difference WENO reconstruction in the Appendix A.2 is used, which is a combination of the interpolation and flux reconstruction, and in this way we will not subject to time step restrictions. We note here this 5th order WENO scheme achieves 5th order when
Δt=CFLΔxmaxx|a(u)(x,t)|. | (40) |
Here "CFL" refers to the CFL number as used in an Eulerian scheme.
First we consider the following linear equation with
ut+ux=0,x∈(0,2). | (41) |
We take the initial condition
For this example, the characteristic equation (7) can be solved exactly, no temporal errors exist. This example is used to test the spatial orders. In Table 1, we show the errors in
Order | Order | |||
20 | 1.25E-4 | – | 2.07E-4 | – |
40 | 3.83E-6 | 5.03 | 7.8E-6 | 4.73 |
80 | 1.15E-7 | 5.06 | 2.38E-7 | 5.04 |
160 | 3.55E-9 | 5.00 | 7.29E-9 | 5.03 |
320 | 1.10E-10 | 5.01 | 2.00E-10 | 5.19 |
640 | 3.42E-12 | 5.01 | 6.03E-12 | 5.05 |
In this example, we consider the quasilinear hyperbolic equation with a variable coefficient
ut+(sin(x)u)x=0,x∈(0,2π). | (42) |
If we take the initial condition
u(x,t)=sin(2arctan(e−ttan(x2)))sin(x). | (43) |
For this example, the RKEI method for solving the characteristic equation (7) becomes the classic RK method. We consider both the 3rd order RKEI scheme with Butcher tableau (2) and the 4th order RKEI scheme with Butcher tableau (3).
For this quasilinear case, a large CFL number is still allowed. We take
3rd order Scheme | 4th order scheme | |||||||
Order | Order | Order | Order | |||||
80 | 6.98E-4 | – | 4.25E-3 | – | 9.99E-5 | – | 1.30E-3 | – |
160 | 9.11E-5 | 2.94 | 6.15E-4 | 2.79 | 6.07E-6 | 4.04 | 2.00E-4 | 2.71 |
320 | 1.17E-5 | 2.96 | 7.92E-5 | 2.96 | 2.81E-7 | 4.43 | 1.62E-5 | 3.62 |
640 | 1.53E-6 | 2.94 | 1.08E-5 | 2.88 | 2.28E-8 | 3.63 | 1.27E-6 | 3.67 |
1280 | 1.95E-7 | 2.97 | 1.69E-6 | 2.95 | 3.09E-9 | 2.88 | 6.38E-8 | 4.32 |
2560 | 2.47E-8 | 2.98 | 1.77E-7 | 2.98 | 4.46E-10 | 2.79 | 7.59E-9 | 3.07 |
For this problem, we also try to test the temporal orders. We take a very refined mesh size
3rd order RKEI (2) | 4th order RKEI (3) | |||||||
Order | Order | Order | Order | |||||
40 | 1.54E-5 | – | 2.39E-5 | – | 5.95E-8 | – | 8.80E-8 | – |
39 | 1.39E-5 | 3.02 | 2.19E-5 | 2.58 | 5.13E-8 | 4.37 | 7.78E-8 | 3.64 |
38 | 1.25E-5 | 3.03 | 1.99E-5 | 2.73 | 4.41E-8 | 4.31 | 6.85E-8 | 3.63 |
37 | 1.12E-5 | 3.02 | 1.74E-5 | 3.69 | 3.88E-8 | 3.52 | 5.76E-8 | 4.77 |
36 | 1.00E-5 | 3.00 | 1.57E-5 | 2.73 | 3.29E-8 | 4.37 | 5.03E-8 | 3.59 |
In the third example, we consider the nonlinear hyperbolic equation. We simply choose the Burger's equation
ut+(u22)x=0,x∈(0,2). | (44) |
We take the initial condition
We first compute the solution up to
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 6.40E-5 | – | 1.36E-4 | – | 2.24E-4 | – |
80 | 7.42E-6 | 3.11 | 1.03E-5 | 3.72 | 2.30E-5 | 3.28 |
160 | 9.57E-7 | 2.95 | 9.29E-7 | 3.47 | 2.58E-6 | 3.16 |
320 | 1.35E-7 | 2.83 | 1.04E-7 | 3.16 | 3.24E-7 | 2.99 |
640 | 1.73E-8 | 2.97 | 1.17E-8 | 3.15 | 3.93E-8 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 3.35E-4 | – | 9.72E-4 | – | 1.20E-3 | – |
80 | 4.75E-5 | 2.82 | 7.51E-5 | 3.69 | 1.30E-4 | 3.26 |
160 | 6.17E-6 | 2.95 | 6.16E-6 | 3.61 | 1.59E-5 | 3.03 |
320 | 8.72E-7 | 2.82 | 7.09E-7 | 3.12 | 2.19E-6 | 2.86 |
640 | 1.10E-7 | 2.98 | 7.64E-8 | 3.22 | 2.66E-7 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.19E-6 | – | 1.12E-4 | – | 1.12E-4 | – |
80 | 2.41E-7 | 4.43 | 7.04E-6 | 3.99 | 6.64E-6 | 4.08 |
160 | 1.09E-8 | 4.47 | 4.67E-7 | 3.91 | 3.95E-7 | 4.07 |
320 | 5.96E-10 | 4.19 | 4.16E-8 | 3.49 | 2.97E-8 | 3.74 |
640 | 3.23E-11 | 4.21 | 4.06E-9 | 3.36 | 2.29E-9 | 3.69 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.66E-5 | – | 9.66E-4 | – | 1.00E-3 | – |
80 | 2.80E-6 | 4.34 | 6.27E-5 | 3.95 | 6.33E-5 | 3.99 |
160 | 8.62E-8 | 5.02 | 3.94E-6 | 3.99 | 3.82E-6 | 4.05 |
320 | 4.01E-9 | 4.42 | 3.43E-7 | 3.52 | 3.16E-7 | 3.59 |
640 | 2.82E-10 | 3.83 | 2.79E-8 | 3.62 | 2.38E-8 | 3.73 |
Then we would like to verify that the three schemes coupled with RKEI, can achieve high order in time for this nonlinear problem. In Table 8, we show the errors and orders for the nonconservative SLFDn scheme with 3rd order RKEI (2) and 4th order RKEI (3), relatively large CFL numbers from
CFL | 3rd order SLFDn | 4th order SLFDn | ||||||
Order | Order | Order | Order | |||||
3 | 4.51E-8 | – | 9.66E-8 | – | 2.33E-10 | – | 2.01E-9 | – |
2.9 | 4.07E-8 | 3.03 | 8.72E-8 | 3.02 | 2.03E-10 | 4.07 | 1.74E-9 | 4.23 |
2.8 | 3.72E-8 | 2.56 | 7.93E-8 | 2.71 | 1.88E-10 | 2.19 | 1.65E-9 | 1.52 |
2.7 | 3.32E-8 | 3.13 | 7.09E-8 | 3.08 | 1.59E-10 | 4.61 | 1.39E-9 | 4.54 |
2.6 | 2.94E-8 | 3.22 | 6.30E-8 | 3.13 | 1.33E-10 | 4.73 | 1.15E-9 | 5.02 |
We now compare the errors and orders for the two conservatives schemes, with 3rd order RKEI (2) and 4th order RKEI (3), in Table 9 and Table 10, respectively. We take CFL numbers from
CFL | 3rd order SLFDc2 | 3rd order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
2 | 4.67E-8 | – | 3.20E-7 | – | 3.07E-8 | – | 1.28E-7 | – |
1.9 | 4.08E-8 | 2.63 | 2.84E-7 | 2.33 | 2.78E-8 | 1.93 | 1.16E-7 | 1.92 |
1.8 | 3.50E-8 | 2.84 | 2.45E-7 | 2.73 | 2.45E-8 | 2.33 | 9.83E-8 | 3.06 |
1.7 | 2.95E-8 | 2.99 | 2.05E-7 | 3.12 | 2.21E-8 | 1.81 | 8.98E-8 | 1.58 |
1.6 | 2.44E-8 | 3.13 | 1.67E-7 | 3.38 | 1.99E-8 | 1.73 | 8.21E-8 | 1.48 |
CFL | 4th order SLFDc2 | 4th order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
3 | 5.46E-10 | – | 2.15E-9 | – | 2.53E-10 | – | 7.94E-10 | – |
2.9 | 4.75E-10 | 4.11 | 1.70E-9 | 6.93 | 2.37E-10 | 1.92 | 6.58E-10 | 5.54 |
2.8 | 4.16E-10 | 3.78 | 1.52E-9 | 3.19 | 2.32E-10 | 0.61 | 5.65E-10 | 4.34 |
2.7 | 3.66E-10 | 3.52 | 1.29E-9 | 4.51 | 2.23E-10 | 1.09 | 4.51E-10 | 6.20 |
2.6 | 3.25E-10 | 3.15 | 1.08E-9 | 4.71 | 2.15E-10 | 0.97 | 3.44E-10 | 7.17 |
Finally we compute the solution to
We further consider another initial condition
Remark 1. For nonlinear hyperbolic problems, large time steps might be beneficial when the dominant time scales are much larger than the characteristic-based time scales [11]. However, when discontinuous solutions exist, such as shock waves are formed, large time steps might still be allowed in some circumstances, e.g., if you track the characteristics very accurately to get an entropy satisfying solution and use total variation bounded (TVB) reconstructions [22]. Here we simply take a relatively large
In this paper, we propose two conservative SLFD schemes for nonlinear scalar hyperbolic conservation laws, which is a combination of the conservative SLFD scheme developed for the quasilinear equation and RKEI scheme for nonlinear ODEs. The resulting schemes are conservative, allow large CFL numbers and can deal with the nonlinear equation. In our numerical tests, we have verified that the conservative schemes can achieve the same error levels as the nonconservative one. 3rd and 4th order in time can be obtained. However, for shock wave solution, the conservative schemes are more robust in shock capturing and avoid the actionless point
Another interesting direction would be applying the conservative SLFD schemes for hyperbolic systems, e.g., the shallow water equations. However, current SL schemes for hyperbolic systems are, e.g., either under the Lagrangian framework which is similar to Lagrangian schemes [28], or updating the Riemann invariants along characteristics [33]. However, the two approaches both are nonconservative and they resort to Eulerian approach for mass conservation. So applying our current conservative SLFD schemes to hyperbolic system is highly nontrivial, and we leave it to our future investigation.
In this appendix, for completeness, we recall the finite difference WENO reconstruction in [19]. After we solve the characteristic equation (7) to get
R=R2∘R1. | (A.1) |
We note that it is important to combine these two reconstructions, especially to achieve large time step without subject to stability restrictions.
For simplicity, we denote the characteristic speed in (7) simply as
We split it into two cases,
Case 1.
ˆhi+12=h−i+12=(xi−x∗i)uni≈Hi=∫xix∗iu(x,tn)dx; | (A.2) |
if
ˆhi+12=h+i+12=(xi−x∗i)uni+1≈Hi=∫xix∗iu(x,tn)dx. | (A.3) |
Case 2.
when
ˆhi+12=i∑j=i∗+1unjΔx+(xi∗−x∗i)uni∗≈Hi=∫xix∗iu(x,tn)dx; | (A.4) |
when
ˆhi+12=i∑ji∗+1unjΔx+(xi∗−x∗i)uni∗≈Hi=∫xixi∗u(x,tn)dx. | (A.5) |
We only consider
First, we consider
S1={uni−2,uni−1,uni},S2={uni−1,uni,uni+1},S3={uni,uni+1,uni+2}. | (A.6) |
The reconstruction
1. reconstruction
h(1)(xi+12)= Δx((1/3ξ−2+1/4ξ2−2+1/18ξ3−2−7/24ξ2−1−7/36ξ3−1−11/24ξ20 +11/36ξ30)uni−2+(−1/3ξ2−2−1/9ξ3−2−7/6ξ−1+7/18ξ3−1 +11/6ξ20−11/18ξ30)uni−1+(1/12ξ2−2+1/18ξ3−2+7/24ξ2−1 −7/36ξ3−1+11/6ξ0−11/8ξ20+11/36ξ30)uni), |
h(2)(xi+12)= Δx((−1/6ξ−1−1/8ξ2−1−1/36ξ3−1+5/24ξ20+5/36ξ30−1/12ξ21 +1/18ξ31)uni−1+(1/6ξ2−1+1/18ξ3−1+5/6ξ0−5/18ξ30+1/3ξ21 −1/9ξ31)uni+(−1/24ξ2−1−1/36ξ3−1−5/24ξ20+5/36ξ30+1/3ξ1 −1/4ξ21+1/18ξ31)uni+1), |
h(3)(xi+12)= Δx((1/3ξ0+1/4ξ20+1/18ξ30+5/24ξ21+5/36ξ31+1/24ξ22 −1/36ξ32)uni+(−1/3ξ20−1/9ξ30+5/6ξ1−5/18ξ31−1/6ξ22 +1/18ξ32)uni+1+(1/12ξ20+1/18ξ30−5/24ξ21+5/36ξ31−1/6ξ2 +1/8ξ22−1/36ξ32)uni+2), |
where
2. compute the linear weights if
γ1=1/10+3/20ξ+1/20ξ3,γ2=3/5+1/10ξ−1/10ξ3,γ3=3/10+1/4ξ+1/20ξ3, |
in this case, all
However when
γ1=1/6,γ2=2/3,γ3=1/6, |
which are used to ensure the non-oscillatory performance [24].
3. compute the smoothness indicator:
β1=13/12(uni−2−2uni−1+uni)2+1/4(uni−2−4uni−1+3uni)2,β2=13/12(uni−1−2uni+uni+1)2+1/4(uni−1−uni+1)2,β3=13/12(uni−2uni+1+uni+2)2+1/4(3uni−4uni+1+uni+2)2, |
4. compute the nonlinear weights.
˜wj=γj(ε+βj)2,wj=˜wj3∑i=1˜wi,j=1,2,3. |
5. get the numerical flux.
ˆhi+12=h−i+12=w1h(1)(xi+12)+w2h(2)(xi+12)+w3h(3)(xi+12). |
When
ˆhi+12=i∑j=i∗+1△xunj+R(ui∗−p,...,ui∗−q)=i∑j=i∗+1Δxunj+ˆhi∗+12, |
where
1. | Rihui Lan, Jingwei Li, Yongyong Cai, Lili Ju, Operator splitting based structure-preserving numerical schemes for the mass-conserving convective Allen-Cahn equation, 2023, 472, 00219991, 111695, 10.1016/j.jcp.2022.111695 |
Algorithm 1: The commutator-free (CF) RKEI method [7]. |
for end |
- |
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- |
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Order | Order | |||
20 | 1.25E-4 | – | 2.07E-4 | – |
40 | 3.83E-6 | 5.03 | 7.8E-6 | 4.73 |
80 | 1.15E-7 | 5.06 | 2.38E-7 | 5.04 |
160 | 3.55E-9 | 5.00 | 7.29E-9 | 5.03 |
320 | 1.10E-10 | 5.01 | 2.00E-10 | 5.19 |
640 | 3.42E-12 | 5.01 | 6.03E-12 | 5.05 |
3rd order Scheme | 4th order scheme | |||||||
Order | Order | Order | Order | |||||
80 | 6.98E-4 | – | 4.25E-3 | – | 9.99E-5 | – | 1.30E-3 | – |
160 | 9.11E-5 | 2.94 | 6.15E-4 | 2.79 | 6.07E-6 | 4.04 | 2.00E-4 | 2.71 |
320 | 1.17E-5 | 2.96 | 7.92E-5 | 2.96 | 2.81E-7 | 4.43 | 1.62E-5 | 3.62 |
640 | 1.53E-6 | 2.94 | 1.08E-5 | 2.88 | 2.28E-8 | 3.63 | 1.27E-6 | 3.67 |
1280 | 1.95E-7 | 2.97 | 1.69E-6 | 2.95 | 3.09E-9 | 2.88 | 6.38E-8 | 4.32 |
2560 | 2.47E-8 | 2.98 | 1.77E-7 | 2.98 | 4.46E-10 | 2.79 | 7.59E-9 | 3.07 |
3rd order RKEI (2) | 4th order RKEI (3) | |||||||
Order | Order | Order | Order | |||||
40 | 1.54E-5 | – | 2.39E-5 | – | 5.95E-8 | – | 8.80E-8 | – |
39 | 1.39E-5 | 3.02 | 2.19E-5 | 2.58 | 5.13E-8 | 4.37 | 7.78E-8 | 3.64 |
38 | 1.25E-5 | 3.03 | 1.99E-5 | 2.73 | 4.41E-8 | 4.31 | 6.85E-8 | 3.63 |
37 | 1.12E-5 | 3.02 | 1.74E-5 | 3.69 | 3.88E-8 | 3.52 | 5.76E-8 | 4.77 |
36 | 1.00E-5 | 3.00 | 1.57E-5 | 2.73 | 3.29E-8 | 4.37 | 5.03E-8 | 3.59 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 6.40E-5 | – | 1.36E-4 | – | 2.24E-4 | – |
80 | 7.42E-6 | 3.11 | 1.03E-5 | 3.72 | 2.30E-5 | 3.28 |
160 | 9.57E-7 | 2.95 | 9.29E-7 | 3.47 | 2.58E-6 | 3.16 |
320 | 1.35E-7 | 2.83 | 1.04E-7 | 3.16 | 3.24E-7 | 2.99 |
640 | 1.73E-8 | 2.97 | 1.17E-8 | 3.15 | 3.93E-8 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 3.35E-4 | – | 9.72E-4 | – | 1.20E-3 | – |
80 | 4.75E-5 | 2.82 | 7.51E-5 | 3.69 | 1.30E-4 | 3.26 |
160 | 6.17E-6 | 2.95 | 6.16E-6 | 3.61 | 1.59E-5 | 3.03 |
320 | 8.72E-7 | 2.82 | 7.09E-7 | 3.12 | 2.19E-6 | 2.86 |
640 | 1.10E-7 | 2.98 | 7.64E-8 | 3.22 | 2.66E-7 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.19E-6 | – | 1.12E-4 | – | 1.12E-4 | – |
80 | 2.41E-7 | 4.43 | 7.04E-6 | 3.99 | 6.64E-6 | 4.08 |
160 | 1.09E-8 | 4.47 | 4.67E-7 | 3.91 | 3.95E-7 | 4.07 |
320 | 5.96E-10 | 4.19 | 4.16E-8 | 3.49 | 2.97E-8 | 3.74 |
640 | 3.23E-11 | 4.21 | 4.06E-9 | 3.36 | 2.29E-9 | 3.69 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.66E-5 | – | 9.66E-4 | – | 1.00E-3 | – |
80 | 2.80E-6 | 4.34 | 6.27E-5 | 3.95 | 6.33E-5 | 3.99 |
160 | 8.62E-8 | 5.02 | 3.94E-6 | 3.99 | 3.82E-6 | 4.05 |
320 | 4.01E-9 | 4.42 | 3.43E-7 | 3.52 | 3.16E-7 | 3.59 |
640 | 2.82E-10 | 3.83 | 2.79E-8 | 3.62 | 2.38E-8 | 3.73 |
CFL | 3rd order SLFDn | 4th order SLFDn | ||||||
Order | Order | Order | Order | |||||
3 | 4.51E-8 | – | 9.66E-8 | – | 2.33E-10 | – | 2.01E-9 | – |
2.9 | 4.07E-8 | 3.03 | 8.72E-8 | 3.02 | 2.03E-10 | 4.07 | 1.74E-9 | 4.23 |
2.8 | 3.72E-8 | 2.56 | 7.93E-8 | 2.71 | 1.88E-10 | 2.19 | 1.65E-9 | 1.52 |
2.7 | 3.32E-8 | 3.13 | 7.09E-8 | 3.08 | 1.59E-10 | 4.61 | 1.39E-9 | 4.54 |
2.6 | 2.94E-8 | 3.22 | 6.30E-8 | 3.13 | 1.33E-10 | 4.73 | 1.15E-9 | 5.02 |
CFL | 3rd order SLFDc2 | 3rd order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
2 | 4.67E-8 | – | 3.20E-7 | – | 3.07E-8 | – | 1.28E-7 | – |
1.9 | 4.08E-8 | 2.63 | 2.84E-7 | 2.33 | 2.78E-8 | 1.93 | 1.16E-7 | 1.92 |
1.8 | 3.50E-8 | 2.84 | 2.45E-7 | 2.73 | 2.45E-8 | 2.33 | 9.83E-8 | 3.06 |
1.7 | 2.95E-8 | 2.99 | 2.05E-7 | 3.12 | 2.21E-8 | 1.81 | 8.98E-8 | 1.58 |
1.6 | 2.44E-8 | 3.13 | 1.67E-7 | 3.38 | 1.99E-8 | 1.73 | 8.21E-8 | 1.48 |
CFL | 4th order SLFDc2 | 4th order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
3 | 5.46E-10 | – | 2.15E-9 | – | 2.53E-10 | – | 7.94E-10 | – |
2.9 | 4.75E-10 | 4.11 | 1.70E-9 | 6.93 | 2.37E-10 | 1.92 | 6.58E-10 | 5.54 |
2.8 | 4.16E-10 | 3.78 | 1.52E-9 | 3.19 | 2.32E-10 | 0.61 | 5.65E-10 | 4.34 |
2.7 | 3.66E-10 | 3.52 | 1.29E-9 | 4.51 | 2.23E-10 | 1.09 | 4.51E-10 | 6.20 |
2.6 | 3.25E-10 | 3.15 | 1.08E-9 | 4.71 | 2.15E-10 | 0.97 | 3.44E-10 | 7.17 |
Algorithm 1: The commutator-free (CF) RKEI method [7]. |
for end |
- |
|||
- |
||||
Order | Order | |||
20 | 1.25E-4 | – | 2.07E-4 | – |
40 | 3.83E-6 | 5.03 | 7.8E-6 | 4.73 |
80 | 1.15E-7 | 5.06 | 2.38E-7 | 5.04 |
160 | 3.55E-9 | 5.00 | 7.29E-9 | 5.03 |
320 | 1.10E-10 | 5.01 | 2.00E-10 | 5.19 |
640 | 3.42E-12 | 5.01 | 6.03E-12 | 5.05 |
3rd order Scheme | 4th order scheme | |||||||
Order | Order | Order | Order | |||||
80 | 6.98E-4 | – | 4.25E-3 | – | 9.99E-5 | – | 1.30E-3 | – |
160 | 9.11E-5 | 2.94 | 6.15E-4 | 2.79 | 6.07E-6 | 4.04 | 2.00E-4 | 2.71 |
320 | 1.17E-5 | 2.96 | 7.92E-5 | 2.96 | 2.81E-7 | 4.43 | 1.62E-5 | 3.62 |
640 | 1.53E-6 | 2.94 | 1.08E-5 | 2.88 | 2.28E-8 | 3.63 | 1.27E-6 | 3.67 |
1280 | 1.95E-7 | 2.97 | 1.69E-6 | 2.95 | 3.09E-9 | 2.88 | 6.38E-8 | 4.32 |
2560 | 2.47E-8 | 2.98 | 1.77E-7 | 2.98 | 4.46E-10 | 2.79 | 7.59E-9 | 3.07 |
3rd order RKEI (2) | 4th order RKEI (3) | |||||||
Order | Order | Order | Order | |||||
40 | 1.54E-5 | – | 2.39E-5 | – | 5.95E-8 | – | 8.80E-8 | – |
39 | 1.39E-5 | 3.02 | 2.19E-5 | 2.58 | 5.13E-8 | 4.37 | 7.78E-8 | 3.64 |
38 | 1.25E-5 | 3.03 | 1.99E-5 | 2.73 | 4.41E-8 | 4.31 | 6.85E-8 | 3.63 |
37 | 1.12E-5 | 3.02 | 1.74E-5 | 3.69 | 3.88E-8 | 3.52 | 5.76E-8 | 4.77 |
36 | 1.00E-5 | 3.00 | 1.57E-5 | 2.73 | 3.29E-8 | 4.37 | 5.03E-8 | 3.59 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 6.40E-5 | – | 1.36E-4 | – | 2.24E-4 | – |
80 | 7.42E-6 | 3.11 | 1.03E-5 | 3.72 | 2.30E-5 | 3.28 |
160 | 9.57E-7 | 2.95 | 9.29E-7 | 3.47 | 2.58E-6 | 3.16 |
320 | 1.35E-7 | 2.83 | 1.04E-7 | 3.16 | 3.24E-7 | 2.99 |
640 | 1.73E-8 | 2.97 | 1.17E-8 | 3.15 | 3.93E-8 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 3.35E-4 | – | 9.72E-4 | – | 1.20E-3 | – |
80 | 4.75E-5 | 2.82 | 7.51E-5 | 3.69 | 1.30E-4 | 3.26 |
160 | 6.17E-6 | 2.95 | 6.16E-6 | 3.61 | 1.59E-5 | 3.03 |
320 | 8.72E-7 | 2.82 | 7.09E-7 | 3.12 | 2.19E-6 | 2.86 |
640 | 1.10E-7 | 2.98 | 7.64E-8 | 3.22 | 2.66E-7 | 3.04 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.19E-6 | – | 1.12E-4 | – | 1.12E-4 | – |
80 | 2.41E-7 | 4.43 | 7.04E-6 | 3.99 | 6.64E-6 | 4.08 |
160 | 1.09E-8 | 4.47 | 4.67E-7 | 3.91 | 3.95E-7 | 4.07 |
320 | 5.96E-10 | 4.19 | 4.16E-8 | 3.49 | 2.97E-8 | 3.74 |
640 | 3.23E-11 | 4.21 | 4.06E-9 | 3.36 | 2.29E-9 | 3.69 |
SLFDn | SLFDc1 | SLFDc2 | ||||
Order | Order | Order | ||||
40 | 5.66E-5 | – | 9.66E-4 | – | 1.00E-3 | – |
80 | 2.80E-6 | 4.34 | 6.27E-5 | 3.95 | 6.33E-5 | 3.99 |
160 | 8.62E-8 | 5.02 | 3.94E-6 | 3.99 | 3.82E-6 | 4.05 |
320 | 4.01E-9 | 4.42 | 3.43E-7 | 3.52 | 3.16E-7 | 3.59 |
640 | 2.82E-10 | 3.83 | 2.79E-8 | 3.62 | 2.38E-8 | 3.73 |
CFL | 3rd order SLFDn | 4th order SLFDn | ||||||
Order | Order | Order | Order | |||||
3 | 4.51E-8 | – | 9.66E-8 | – | 2.33E-10 | – | 2.01E-9 | – |
2.9 | 4.07E-8 | 3.03 | 8.72E-8 | 3.02 | 2.03E-10 | 4.07 | 1.74E-9 | 4.23 |
2.8 | 3.72E-8 | 2.56 | 7.93E-8 | 2.71 | 1.88E-10 | 2.19 | 1.65E-9 | 1.52 |
2.7 | 3.32E-8 | 3.13 | 7.09E-8 | 3.08 | 1.59E-10 | 4.61 | 1.39E-9 | 4.54 |
2.6 | 2.94E-8 | 3.22 | 6.30E-8 | 3.13 | 1.33E-10 | 4.73 | 1.15E-9 | 5.02 |
CFL | 3rd order SLFDc2 | 3rd order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
2 | 4.67E-8 | – | 3.20E-7 | – | 3.07E-8 | – | 1.28E-7 | – |
1.9 | 4.08E-8 | 2.63 | 2.84E-7 | 2.33 | 2.78E-8 | 1.93 | 1.16E-7 | 1.92 |
1.8 | 3.50E-8 | 2.84 | 2.45E-7 | 2.73 | 2.45E-8 | 2.33 | 9.83E-8 | 3.06 |
1.7 | 2.95E-8 | 2.99 | 2.05E-7 | 3.12 | 2.21E-8 | 1.81 | 8.98E-8 | 1.58 |
1.6 | 2.44E-8 | 3.13 | 1.67E-7 | 3.38 | 1.99E-8 | 1.73 | 8.21E-8 | 1.48 |
CFL | 4th order SLFDc2 | 4th order SLFDc1 | ||||||
Order | Order | Order | Order | |||||
3 | 5.46E-10 | – | 2.15E-9 | – | 2.53E-10 | – | 7.94E-10 | – |
2.9 | 4.75E-10 | 4.11 | 1.70E-9 | 6.93 | 2.37E-10 | 1.92 | 6.58E-10 | 5.54 |
2.8 | 4.16E-10 | 3.78 | 1.52E-9 | 3.19 | 2.32E-10 | 0.61 | 5.65E-10 | 4.34 |
2.7 | 3.66E-10 | 3.52 | 1.29E-9 | 4.51 | 2.23E-10 | 1.09 | 4.51E-10 | 6.20 |
2.6 | 3.25E-10 | 3.15 | 1.08E-9 | 4.71 | 2.15E-10 | 0.97 | 3.44E-10 | 7.17 |