A priori error estimates of Adams-Bashforth discontinuous Galerkin methods for scalar nonlinear conservation laws
Charles Puelz Beatrice Riviere



A priori error estimates of Adams–Bashforth  
 discontinuous Galerkin methods for scalar nonlinear conservation laws
    Charles Puelz and Béatrice Rivière
    December 30, 2023
====================================================================================================================



In this paper we show theoretical convergence of a second–order Adams–Bashforth discontinuous Galerkin method for approximating smooth solutions to scalar nonlinear conservation laws with E-fluxes.  A priori error estimates are also derived for a first–order forward Euler discontinuous Galerkin method.   Rates are optimal in time and suboptimal in space; they are valid under a CFL condition.



§ INTRODUCTION


We consider approximating smooth solutions to the following nonlinear partial differential equation posed with initial conditions:

    ∂ u/∂ t + ∂/∂ xf(u) = s(u),   in ℝ× (0,T],
       u = u_0,   in ∈ℝ×{0},

where u:ℝ× [0,T] →ℝ and f,s:ℝ→ℝ. The function s is assumed to be Lipschitz. As typical for the numerical analysis of such problems <cit.>, we do not consider boundary conditions, and instead assume the solution has compact support in some interval [0,L].  


The focus of this work is the analysis of  the second order Adams–Bashforth method in time combined with the discontinuous Galerkin  method in space.  The main motivation for studying this discretization is its popularity in the hemodynamic modeling community for approximating a nonlinear hyperbolic system describing blood flow in an elastic vessel <cit.>.  For a selection of work simulating this model with a discontinuous Galerkin spatial discretization coupled to the second order Adams–Bashforth scheme, see  <cit.>.  To the best of our knowledge, there is little analysis for this fully discrete scheme.  The results presented in this paper for scalar hyperbolic equations provide a first step towards theoretically understanding the numerical approximation of the hyperbolic system modeling blood flow.
In addition, we provide an error analysis for the first order forward Euler in time combined with discontinuous Galerkin in space.


Discontinuous Galerkin schemes for hyperbolic conservations laws have been extensively studied, especially when coupled with Runge–Kutta methods for the time discretization.  This class of schemes was introduced in the series of papers by Cockburn, Shu, and co-authors <cit.>.  We recall the work from Zhang, Shu, and others analyzing Runge–Kutta discontinuous Galerkin methods applied to scalar conservation laws and symmetrizable systems <cit.>.  These papers establish error estimates for smooth solutions for both second and third order Runge–Kutta schemes.  Their analysis requires the CFL condition Δ t = O(h^4/3) for the second order Runge–Kutta scheme and piecewise polynomials of degree two and higher.  The CFL condition Δ t = O(h) may be used for the third order Runge-Kutta scheme for piecewise polynomials of degree one and higher and for the second order Runge–Kutta scheme with piecewise linear polynomials.

Recent stability and convergence results have been obtained for IMEX (implicit–explicit) multistep schemes applied to a nonlinear convection diffusion equation, i.e. (<ref>)–(<ref>) augmented with a nonzero diffusion term  <cit.>.  These schemes implicitly discretize the diffusion term and explicitly discretize the hyperbolic term.  It is not immediately clear how to adapt the analysis to the case of zero diffusion since the estimates depend on the reciprocal of the diffusion parameter.  


A summary of the paper is as follows.  In Section <ref>, we introduce the numerical schemes, properties of the numerical flux, and inequalities related to projections.  The main results are also stated.  Section <ref> and  <ref> contain the proofs of the convergence results. In Section <ref> we provide some numerical results for inviscid Burger's equation and a nonlinear hyperbolic system modeling blood flow in an elastic vessel.  Conclusions follow.






§ SCHEME AND MAIN RESULTS



We define notation relevant for the spatial discretization of (<ref>)–(<ref>) by the discontinuous Galerkin method.  To do this, we make a similar technical modification to the flux function as in <cit.>.  If the initial condition u_0 takes values within some open set Ω, then locally in time the solution to (<ref>)–(<ref>) also takes values in Ω <cit.>.  We assume the flux function f ∈ C^3(ℝ) vanishes outside of Ω so derivatives up to third order are uniformly bounded, i.e. there exists some constant C depending only on f and its derivatives satisfying:

    |f^(γ)(v)| ≤ C,   ∀ v ∈ℝ,   γ = 1,2,3.
 

Let the collection of intervals ( I_j )_j=0^N be a uniform partition of the interval [0,L], with I_j = [x_j,x_j+1] of  size h. Let ℙ^k(I_j) denote the space of polynomials of degree k on the interval I_j.  The approximation space is

    𝕍_h={ϕ_h:[0,L] →ℝ s.t. ϕ_h|_I_j∈ℙ^k(I_j),  ∀ j = 0, …, N }.

The space L^2(0,L) is the standard L^2 space; let (·,·) denote the L^2 inner-product over Ω, with associated norm ‖·‖. 
Let Π_h be the L^2 projection into 𝕍_h:

    (Π_h v, ϕ_h) = (v,ϕ_h),  ∀ϕ_h∈𝕍_h,   ∀ v∈ L^2(0,L).

Define the notation for traces of a function ϕ:[0,L]→ℝ to the boundaries of the intervals:

    ϕ^±|_x_j   = lim_ε→ 0,  ε > 0ϕ(x_j ±ε),    1≤ j≤ N, 
    ϕ^+|_x_0   = lim_ε→ 0,  ε > 0ϕ(x_0 + ε), 
    ϕ^-|_x_N+1   = lim_ε→ 0,  ε > 0ϕ(x_N+1 -ε).

The standard notation for jumps and averages at the interior nodes is given as follows:

    [ϕ]|_x_j   = ϕ^-|_x_j - ϕ^+|_x_j,    1≤ j≤ N, 
    {ϕ}|_x_j   = 1/2(ϕ^-|_x_j + ϕ^+|_x_j),    1≤ j≤ N.

Let f̂ denote the numerical flux, that is assumed to be Lipschitz continuous and consistent.

There is a constant C_L > 0 such that for any p, q, u, v ∈ℝ:

    |f̂(p, q) - f̂(u, v)| ≤ C_L ( |p - u| + |q - v| ),

and 

    f̂(v,v) = f(v),   ∀ v ∈ℝ.



We also assume that f̂ belongs to the class of E–fluxes <cit.>. 

The numerical flux f̂ is an E–flux, which means it satisfies, for all w between v^- and v^+, 
	
    ( f̂(v^-,v^+) - f(w) ) [v]|_x_j≥ 0,    1 ≤ j ≤ N.



An example of a numerical flux that satisfies Assumption <ref> and Assumption <ref> is the local Lax-Friedrichs flux, f̂_LF,  defined by:
	
    f̂_LF(v^-, v^+)|_x_j = { f(v) }|_x_j + 1/2J(v^-, v^+) [v]|_x_j,   ∀ 1≤ j ≤ N,

with 	

    J(v^-, v^+)|_x_j = max_min(v^-|_x_j, v^+|_x_j) ≤ w ≤max(v^-|_x_j, v^+|_x_j) |f'(w)| ,   ∀ j = 1, …, N.

Finally, we define a discrete function α at each interior node. The fact that α is nonnegative and
uniformly bounded is a key ingredient in the error analysis.

    α(v)|_x_j =
    		{[ [v]^-1( f̂(v^-, v^+) - f({v}) )|_x_j,                      if  [v]|_x_j≠ 0,;                    1/2|f'({v}|_x_j)|,                     if  [v]|_x_j = 0. ].



There exist constants C_α, C_0 and C_1 such that

    0 ≤α(v)|_x_j   ≤ C_α,   ∀ (v^-, v^+) ∈ℝ^2,   ∀  1≤ j≤ N,
    1/2|f'({v}|_x_j)|    ≤α(v)|_x_j + C_0 |[v]|_x_j|,   ∀ (v^-, v^+) ∈ℝ^2,   ∀ 1≤ j≤ N, 
    1/8 f”({v}|_x_j)[v]|_x_j   ≤α(v)|_x_j + C_1 [v]^2|_x_j,   ∀ (v^-, v^+) ∈ℝ^2,   ∀ 1≤ j≤ N.


The constants C_0 and C_1 depend on the derivatives of f.

The proof of Lemma <ref> follows the one in <cit.>; the definition for α slightly differs 
from the one given in <cit.> so that it is suitable for the error analysis of the
Adams–Bashforth scheme.


An additional assumption is made for the numerical flux. 

There is a constant C > 0 such that for any v_h∈𝕍_h and v∈𝒞(0,L):

    |α(v_h)|_x_j -α(v)|_x_j|≤ C ‖ v_h-v‖_∞,  ∀ 1≤ j≤ N.




Assumption <ref> is used in the error analysis for the Adams–Bashforth scheme.  It is easy to check
that the local Lax-Friedrichs flux defined by (<ref>) satisfies (<ref>).




We now introduce the discontinuous Galerkin discretization on each interval.

    ℋ_j(v, ϕ_h)    = ∫_I_jf(v) d ϕ_h/dx 
    + ∫_I_js(v) ϕ_h -f̂(v^-, v^+)|_x_j+1ϕ_h^-|_x_j+1 
    + f̂(v^-, v^+)|_x_jϕ_h^+|_x_j  ∀  1≤ j≤ N-1, 
    ℋ_0(v, ϕ_h)    = ∫_I_0f(v) d ϕ_h/dx 
    + ∫_I_0s(v) ϕ_h -f̂(v^-, v^+)|_x_1ϕ_h^-|_x_1, 
    ℋ_N(v, ϕ_h)    =∫_I_Nf(v) d ϕ_h/dx 
    + ∫_I_Ns(v) ϕ_h + f̂(v^-, v^+)|_x_Nϕ_h^+|_x_N.

For some number M>0, define Δ t = T / M. 

The second order in time Adams–Bashforth scheme is:
given u_h^0 ∈𝕍_h and u_h^1 ∈𝕍_h,  for n = 1, …, M-1, seek u_h^n+1∈𝕍_h satisfying

    ∫_I_ju_h^n+1ϕ_h  =   ∫_I_j u_h^nϕ_h  + Δ t 3/2ℋ_j(u_h^n, ϕ_h) - Δ t 1/2ℋ_j(u_h^n-1, ϕ_h),
      ∀ϕ_h ∈𝕍_h,   ∀  0≤ j≤ N.

Since (<ref>) is a multi-step method, two starting values are needed.  We choose 
u_h^0 = Π_h u_0  for the initial value, and we choose u_h^1 = ũ_h^1 where
ũ_h^1  satisfies the first-order in time
forward Euler scheme defined below.


With the choice ũ_h^0 = Π_h u_0, for n = 0, …, M-1, seek ũ_h^n+1∈𝕍_h satisfying

    ∫_I_jũ_h^n+1ϕ_h  =   ∫_I_jũ_h^nϕ_h  + Δ t   ℋ_j(ũ_h^n, ϕ_h),  ∀ϕ_h ∈𝕍_h,   ∀  0≤ j≤ N.

The initial value u_h^1 is computed using (<ref>) with a time step that is small enough so that the following
assumption holds:

    u_h^1 - Π_h u^1≤ h^k+1/2.

Theorem <ref> below shows that (<ref>) is a reasonable assumption if the time step used for
the forward Euler method is small enough.

The main result of this paper is the convergence result for the Adams-Bashforth scheme (<ref>).


Assume the exact solution u belongs to 𝒞^2([0,T];H^k+1(Ω)). 
Let u_h^1 satisfy (<ref>). 
Under Assumptions <ref>, <ref>, <ref> and the CFL condition 
Δ t = O(h^2), there is a constant C independent of h and Δ t such that, 
 for h sufficiently small, and for k ≥ 2: 
	
    max_n = 0, …, Mu^n - u_h^n≤ C (Δ t^2 + h^k+1/2).


The proof of Theorem <ref> is given in Section <ref>. An easy modification of the proof yields the
following convergence result for the forward Euler scheme (<ref>). Its proof is outlined in Section <ref>.


Assume the exact solution u belongs to 𝒞^2([0,T];H^k+1(Ω)). 
Let (ũ_h^n)_n satisfy (<ref>).
Under Assumptions <ref>, <ref> and the CFL condition 
Δ t = O(h^2), for h sufficiently small, and 
for k ≥ 1, there is a constant C independent of h and Δ t such that:	

    max_n = 0, …, Mu^n - ũ_h^n≤ C (Δ t + h^k+1/2).



We remark that von Neumann stability analysis conducted in <cit.> suggests a less restrictive CFL condition Δ t = O(h^4/3) for the second order Adams–Bashforth scheme.  Our theoretical estimates require Δ t = O(h^2); at the moment we are unable to relax this condition. 


 
We finish this section by recalling inverse inequalities, trace inequalities and approximations results.
Let v_∞ = max_x ∈ [0,L]|v(x)| denote the sup-norm. There exists a constant C independent of h such that

    ϕ_h_∞   ≤ C h^-1/2ϕ_h,   ∀ϕ_h∈𝕍_h,
    
    |ϕ_h^n,±|_x_j|    ≤ C h^-1/2ϕ_h_L^2(I_j),  ∀  1≤ j≤ N,   ∀ϕ_h∈𝕍_h,
    	
    ( ∑_j = 0^Nd/dxϕ_h_L^2(I_j)^2 )^1/2   ≤ C h^-1ϕ_h,   ∀ϕ_h∈𝕍_h.

For simplicity we denote u^n the function u evaluated at the time t^n = n Δ t. The approximation error is denoted

    η^n  = u^n - Π_h u^n,

and it satisfies the optimal a priori bounds

    η^n   ≤ C h^k+1, 
    
    |η^n,±|_x_j|    ≤ C h^k+1/2,  ∀  1≤ j≤ N,
    
    
    η^n_∞   ≤ C h^k+1/2, 
    η^n+1 - η^n    ≤ C Δ t h^k+1.

The constant C is independent of h, Δ t but depends on the exact solution u and its derivatives.






§ PROOF OF THEOREM <REF>



For the error analysis, we denote

    χ^n = u_h^n-Π_h u^n.

The proof of Theorem <ref> is based on an induction hypothesis:

    χ^ℓ≤ h^3/2,   ∀  0≤ℓ≤ M.

Since χ^0=0, the hypothesis (<ref>) is trivially satisfied for ℓ=0.  
With the assumption (<ref>), it is also true for ℓ=1.
Fix ℓ∈{2,…,M} and assume that 

    χ^n≤ h^3/2,   ∀  0≤ n ≤ℓ-1.

We will show that (<ref>) is valid for n = ℓ.  We begin by deriving an error inequality.
We fix an interval I_j for 0≤ j≤ N. 
It is easy to see that the scheme is consistent in space and the exact solution satisfies

    3/2∫_I_j    u_t^n ϕ_h - 1/2∫_I_ju_t^n-1ϕ_h = 3/2ℋ_j(u^n, ϕ_h) - 1/2ℋ_j(u^n-1, ϕ_h),   ∀ 1≤ n ≤ M-1.

In the above, the notation u_t^n is used for the time derivative of u evaluated at t^n.
Subtracting (<ref>) from (<ref>) and rearranging terms, one obtains:

    ∫_I_j   (u_h^n+1 -u_h^n - Δ t3/2 u_t^n  + Δ t1/2 u_t^n-1)ϕ_h 
       = Δ t3/2(ℋ_j(u_h^n, ϕ_h) - ℋ_j(u^n, ϕ_h) ) - Δ t1/2 (ℋ_j(u_h^n-1, ϕ_h) - ℋ_j(u^n-1, ϕ_h) ),   ∀ 1≤ n ≤ M-1.

Summing over the elements j = 0, …, N and adding and subtracting the L^2 projection 
of u at t^n and t^n+1 yields the equality:

    ∫_0^L (χ^n+1 - χ^n) ϕ_h    =  ∫_0^L (u^n -u^n+1 + Δ t 3/2 u_t^n  - Δ t1/2 u_t^n-1)ϕ_h  +  ∫_0^L (η^n+1 - η^n) ϕ_h  + b^n(ϕ_h),

with the following definition for n ≥ 1 

    b^n(ϕ_h)    = Δ t 3/2∑_j = 0^N ( ℋ_j(u_h^n, ϕ_h) - ℋ_j(u^n, ϕ_h)) - Δ t  1/2∑_j = 0^N ( ℋ_j(u_h^n-1, ϕ_h) - ℋ_j(u^n-1, ϕ_h) ).

The second term on the right hand side of (<ref>) vanishes due to the property (<ref>) of the local L^2 projection.  To handle the first term, we obtain from the following Taylor expansions for some ζ̃∈ [t^n-1,t^n] and some ζ∈ [t^n, t^n+1]:

    u^n+1 - u^n    = Δ t u_t^n  + 1/2Δ t^2 u_tt^n  + 1/6Δ t^3 u_ttt|_ζ, 
    
    	u_t^n-1 - u_t^n     = -Δ t u_tt^n  + 1/2Δ t^2 u_ttt|_ζ̃.

Thus we have

    u^n -u^n+1 + Δ t 3/2 u_t^n  - Δ t1/2 u_t^n-1   = - Δ t^3 (1/6u_ttt|_ζ +1/4u_ttt|_ζ̃).

Hence (<ref>) becomes:

    ∫_0^L (χ^n+1 - χ^n ) ϕ_h  ≤ C Δ t^3 ∫_0^L |ϕ_h| +  b^n(ϕ_h).

Cauchy Schwarz's inequality and Young's inequalities imply:

    ∫_0^L (χ^n+1 - χ^n ) ϕ_h  ≤ C Δ t^5 +  Δ t  ϕ_h^2   + b^n(ϕ_h).

We choose ϕ_h = χ^n in inequality (<ref>) to obtain:

    ∫_0^L (χ^n+1 - χ^n ) χ^n  ≤  C Δ t^5 +  Δ t  χ^n^2  + b^n(χ^n).
 
So, the following error inequality holds for n ≥ 1:

    1/2χ^n+1^2 - 1/2χ^n^2  ≤  
     C Δ t^5 +  Δ t  χ^n^2  
    +1/2χ^n+1 - χ^n^2 
    + b^n(χ^n).

It remains to handle the last two terms in (<ref>).  The proofs of the following two lemma are given in the next section.

 
Assume that Δ t = O(h^2).  The following holds for n ≥ 1:

    χ^n+1 - χ^n^2    ≤ C Δ t^6 + C Δ t   ( χ^n^2 + χ^n-1^2) + C Δ t   h^2k+2.



 
Let n≥ 2 and assume χ^n≤ h^3/2, ‖χ^n-1‖≤ h^3/2,  and Δ t = O(h^2).  The following holds:

    b^n(χ^n) ≤   C Δ t (‖χ^n‖^2+‖χ^n-1‖^2) 
    + C Δ t^6 
    + C Δ t  (1+ 2 ε^-1) h^2k+1
       -(1/2-2 ε) Δ t ∑_j=1^N α(u_h^n)|_x_j [χ^n]^2|_x_j
    -(1/2-2 ε) Δ t ∑_j=1^N α(u_h^n-1)|_x_j [χ^n-1]^2|_x_j,   ∀ε >0.

For n = 1 one has the following:

    b^1(χ^1) ≤   C Δ t ‖χ^1‖^2
    + C Δ t  (1+ 2 ε^-1) h^2k+1
       +3‖χ^1‖^2-(1/2-2 ε) Δ t ∑_j=1^N α(u_h^1)|_x_j [χ^1]^2|_x_j,  ∀ε >0.


Substituting the bounds from (<ref>), (<ref>), (<ref>) (with ε = 1/4), and using the fact that 
α(u_h^n) and α(u_h^n-1) are nonnegative, the error inequality (<ref>) simplifies to:

    χ^n+1^2 - χ^n^2    ≤ C Δ t^5 +  C Δ t   ( χ^n^2 + χ^n-1^2 + ‖χ^n-2‖^2)  + C Δ t h^2k+1,   
    n≥ 2,

and

    χ^n+1^2 - χ^n^2    ≤ C Δ t^5 +  C Δ t  χ^n^2   + C Δ t h^2k+1
    + C ‖χ^n‖^2,   
    n = 1.

Summing (<ref>) from n = 2, …, ℓ-1 and adding to (<ref>) one obtains:

    χ^ℓ^2     ≤ C Δ t^4 + Ch^2k+1 +  Cχ^1^2  +  C Δ t  ∑_n = 0^ℓ-1χ^n^2.

Gronwall's inequality  and assumption (<ref>) immediately gives

    χ^ℓ^2 ≤ C_2 T e^T  ( Δ t^4 +  h^2k+1),

where C_2 is independent of ℓ, h and Δ t.  Employing the CFL condition Δ t = O(h^2), one has:

    χ^ℓ≤(C_2 T e^T )^1/2( h^4 +  h^k+1/2).

The induction proof is complete if h is small enough so that

    C_2 T e^T h < 1,
 
implying that for k ≥ 2:

    χ^ℓ≤(C_2 T e^T )^1/2 h ( h^3 +  h^k-1/2)  ≤ h^3/2.

Since η^n≤ C h^k+1 and u^n - u_h^n≤η^n + χ^n we can conclude:

    u^n - u_h^n≤ C(Δ t^2 + h^k+1/2).
	




 §.§ Proof of Lemma <ref>


Choose ϕ_h = χ^n+1 - χ^n in (<ref>) and use Cauchy-Schwarz's and Young's inequalities to obtain:

    χ^n+1 - χ^n^2 ≤  CΔ t^6 + 2  b^n(χ^n+1 - χ^n).
	
We will now obtain a bound for b(ϕ_h) for any ϕ_h ∈𝕍_h. 
By definition, we write

    b^n(ϕ_h) = θ_1 + θ_2 + θ_3,

where

    θ_1  = 3/2Δ t ∑_j=0^N ∫_I_j (f(u_h^n)-f(u^n))dϕ_h/dx
    -1/2Δ t ∑_j=0^N ∫_I_j (f(u_h^n-1)-f(u^n-1))dϕ_h/dx
    
    -3/2Δ t ∑_j=1^N (f({u_h^n})-f(u^n))|_x_j [ϕ_h]|_x_j
    +1/2Δ t ∑_j=1^N (f({u_h^n-1})-f(u^n-1))|_x_j [ϕ_h]|_x_j,


    θ_2 =  Δ t ∑_j=0^N ∫_I_j(3/2 (s(u_h^n)-s(u^n))-1/2 (s(u_h^n-1)-s(u^n-1)))ϕ_h,


    θ_3 = -3/2Δ t ∑_j=1^N (f̂(u_h^n,-,u_h^n,+)-f({u_h^n}))|_x_j [ϕ_h]|_x_j
    +1/2Δ t ∑_j=1^N (f̂(u_h^n-1,-,u_h^n-1,+)-f({u_h^n-1}))|_x_j [ϕ_h]|_x_j.

Using Taylor expansions, we write for some ζ_1^n, ζ_2^n, ζ_1^n-1 and ζ_2^n-1:

    f(u_h^n) - f(u^n)    = f'(ζ_1^n)(u_h^n - u^n) = f'(ζ_1^n)(χ^n-η^n), 
    
    	f({u_h^n}) - f(u^n)    = f'(ζ_2^n)({u_h^n} - {u^n}) =  f'(ζ_2^n)({χ^n} - {η^n}), 
    
    	f(u_h^n-1) - f(u^n-1)    = f'(ζ_1^n-1)(u_h^n-1 - u^n-1) = f'(ζ_1^n-1)(χ^n-1-η^n-1), 
    
    	f({u_h^n-1}) - f(u^n-1)    = f'(ζ_2^n)({u_h^n-1} - {u^n-1}) =  f'(ζ_2^n-1)({χ^n-1} - {η^n-1}).

Using the above expansions in the definition of θ_1, trace inequalities and the CFL condition Δ t =𝒪(h^2), we
can obtain for any ε > 0

    |θ_1|  ≤ε‖ϕ_h‖^2 + C ε^-1Δ t (‖χ^n‖^2 +‖χ^n-1‖^2)
    +C ε^-1 Δ t   h^2k+2.

The term θ_2 is bounded using Lipschitz continuity of s, approximation results, Cauchy-Schwarz's and Young's inequalities.
For any ε>0, we have

    θ_2 
    ≤ C ε^-1Δ t^2 h^2k+2 + Cε^-1Δ t^2 (χ^n^2+‖χ^n-1‖^2)  
    + εϕ_h^2.

Lastly, the term θ_3 can be rewritten using the definition  (<ref>). 

    θ_3 =     -3/2Δ t ∑_j = 1^N α(u_h^n)|_x_j [u_h^n]|_x_j [ϕ_h]|_x_j 
    +1/2Δ t ∑_j = 1^N α(u_h^n-1)|_x_j [u_h^n-1]|_x_j [ϕ_h]|_x_j
    
    =     -3/2Δ t ∑_j = 1^N α(u_h^n)|_x_j [χ^n-η^n]|_x_j [ϕ_h]|_x_j 
    +1/2Δ t ∑_j = 1^N α(u_h^n-1)|_x_j [χ^n-1-η^n-1]|_x_j [ϕ_h]|_x_j.

Using Young's and Cauchy-Schwarz's inequalities, approximation results, trace inequalities, boundedness of α and the CFL condition, we have

    |θ_3 | ≤ε‖ϕ_h‖^2 
    + C ε^-1Δ t (‖χ^n‖^2+‖χ^n-1‖^2)
    + C ε^-1Δ t   h^2k+2.

Combining the bounds above yields

    b(ϕ_h) ≤ε‖ϕ_h‖^2 +
    C ε^-1Δ t (‖χ^n‖^2+‖χ^n-1‖^2)
    + C ε^-1Δ t   h^2k+2,   ∀ε >0,   ∀ϕ_h∈𝕍_h.

We choose ε = 1/4 and ϕ_h  = χ^n-χ^n-1 in (<ref>) and substitute the bound 
in (<ref>) to obtain (<ref>).

    χ^n+1 - χ^n^2 ≤  CΔ t^6 
    + C  Δ t (‖χ^n‖^2+‖χ^n-1‖^2)
    + C Δ t   h^2k+2.
	




 §.§ Proof of Lemma <ref>



As in the proof of Lemma <ref>, we write 

    b^n(χ^n) = θ_1 + θ_2 + θ_3,

where the definitions of θ_1, θ_2, θ_3 are given in (<ref>), (<ref>) and (<ref>) respectively
for the particular choice ϕ_h = χ^n.  Unfortunately we cannot make use of the bound (<ref>) since the factor
Δ t is missing in front of ε‖ϕ_h ‖^2.  A more careful analysis is needed, and we will take
advantage of the CFL condition.

Define

    ℱ(n,ϕ_h) = 
    Δ t ∑_j=0^N ∫_I_j (f(u_h^n)-f(u^n))dϕ_h/dx
    - Δ t ∑_j=1^N (f({u_h^n})-f(u^n))|_x_j [ϕ_h]|_x_j.

Using the function ℱ which is linear in its second argument, we rewrite the term θ_1 as

    θ_1 = 3/2ℱ(n,χ^n) -1/2ℱ(n-1,χ^n-1) + 1/2ℱ(n-1,χ^n-1-χ^n).

We now state a bound for the term ℱ(n,χ^n). 

    ℱ(n,χ^n)≤ C Δ t ‖χ^n‖^2
    + C(1+ε^-1) Δ t   h^2k+1
    +εΔ t ∑_j=1^N α(u_h^n)|_x_j [χ^n]^2|_x_j,  ∀ε > 0.

The proof of (<ref>) is technical and can be found in Appendix <ref>.
The bound for ℱ(n-1,χ^n-1) is identical.

    ℱ(n-1,χ^n-1)≤ C Δ t ‖χ^n-1‖^2
    + C(1+ε^-1) Δ t   h^2k+1
    +εΔ t ∑_j=1^N α(u_h^n-1)|_x_j [χ^n-1]^2|_x_j,  ∀ε > 0.

We are left with bounding ℱ(n-1,χ^n-1-χ^n). Following the technique used for bound (<ref>), we can obtain

    ℱ(n-1,χ^n-1-χ^n)   ≤‖χ^n-1-χ^n‖^2 
    + C  Δ t ‖χ^n-1‖^2 +C  Δ t   h^2k+2.

Combining the above with (<ref>), we have for n≥ 2

    θ_1 ≤    C Δ t (‖χ^n‖^2+ ‖χ^n-1‖^2 + ‖χ^n-2‖^2)
    + C (1+2 ε^-1) Δ t   h^2k+1
        +εΔ t ∑_j=1^N α(u_h^n)|_x_j [χ^n]^2|_x_j
     +εΔ t ∑_j=1^N α(u_h^n-1)|_x_j [χ^n-1]^2|_x_j
    + C Δ t^6,   ∀ϵ>0.

For n=1, since χ^0 = 0, inequalities (<ref>) and (<ref>) imply

    θ_1 ≤   C Δ t ‖χ^1‖^2
    + C (1+ε^-1) Δ t   h^2k+1
      +εΔ t ∑_j=1^N α(u_h^1)|_x_j [χ^1]^2|_x_j
    + ‖χ^1‖^2,
      ∀ϵ>0.






The term θ_2 is bounded using Lipschitz continuity of s, approximation results, Cauchy-Schwarz's
inequality:

    θ_2 ≤ C Δ t (‖χ^n‖+‖η^n‖+‖χ^n-1‖+‖η^n-1‖) ‖χ^n‖≤ C Δ t (‖χ^n‖^2 +‖χ^n-1‖^2) + C Δ t   h^2k+2.


For the term θ_3, we use the definition (<ref>) and write

    θ_3 = -3/2Δ t ∑_j = 1^N α(u_h^n)|_x_j [χ^n-η^n]|_x_j [χ^n]|_x_j 
    +1/2Δ t ∑_j = 1^N α(u_h^n-1)|_x_j [χ^n-1-η^n-1]|_x_j [χ^n]|_x_j.

After some manipulation we rewrite θ_3 as:

    θ_3 = -1/2Δ t ∑_j = 1^N α(u_h^n)|_x_j [χ^n]^2|_x_j
    -1/2Δ t ∑_j = 1^N α(u_h^n-1)|_x_j [χ^n-1]^2|_x_j
    
    +Δ t ∑_j=1^N (α(u_h^n-1)-α(u_h^n))|_x_j [χ^n-1-η^n-1]|_x_j [χ^n-1]|_x_j
    -1/2Δ t ∑_j=1^N  α(u_h^n-1)|_x_j [χ^n-1-η^n-1]|_x_j [χ^n-1-χ^n]|_x_j
    
    +Δ t ∑_j=1^N  α(u_h^n)|_x_j [χ^n-1-η^n-1]|_x_j [χ^n-1-χ^n]|_x_j
    +Δ t ∑_j=1^N  α(u_h^n)|_x_j [(χ^n-1-χ^n)-(η^n-1-η^n)]|_x_j [χ^n]|_x_j
    
    +1/2Δ t ∑_j=1^N  α(u_h^n)|_x_j [η^n]|_x_j [χ^n]|_x_j 
    +1/2Δ t ∑_j=1^N  α(u_h^n-1)|_x_j [η^n-1]|_x_j [χ^n-1]|_x_j.



We now bound the terms in the right-hand side of (<ref>) except for the first two terms.
We write

    α(u_h^n-1)|_x_j - α(u_h^n)|_x_j
    =(α(u_h^n-1)|_x_j - α(u^n-1)|_x_j)
    +(α(u^n-1)|_x_j - α(u^n)|_x_j)
    -(α(u_h^n)|_x_j-α(u^n)|_x_j).

From (<ref>) and (<ref>), we have

    |α(u_h^n-1)|_x_j - α(u_h^n)|_x_j|
    ≤ C‖ u_h^n-1 - u^n-1‖_∞ + C ‖ u_h^n-u^n‖_∞
    +1/2|   | f'(u^n-1)|_x_j| -| f'(u^n)|_x_j| |.

With a Taylor expansion, we obtain

    | α(u_h^n-1)|_x_j   - α(u_h^n)|_x_j| ≤  C (u^n-1 - u_h^n-1_∞ + u^n - u_h^n_∞ + Δ t ),   ∀ 1≤ j≤ N.

With the assumption ‖χ^n‖≤ h^3/2 and ‖χ^n-1‖≤ h^3/2,  bound (<ref>)
and approximation results, we have 

    | α(u_h^n-1)|_x_j   - α(u_h^n)|_x_j| ≤  C (h + Δ t),   ∀ 1≤ j ≤ N.

Using trace inequalities, we then have

    Δ t ∑_j=1^N (α(u_h^n-1)|_x_j-α(u_h^n)|_x_j)[χ^n-1]^2|_x_j≤ C Δ t (1 + h^-1Δ t) ‖χ^n-1‖^2.

With the CFL condition, we conclude

    Δ t ∑_j=1^N (α(u_h^n-1)|_x_j-α(u_h^n)|_x_j)[χ^n-1]^2|_x_j≤ C Δ t ‖χ^n-1‖^2.

Similarly we have

    -Δ t ∑_j=1^N (α(u_h^n-1)|_x_j-α(u_h^n)|_x_j)[η^n-1]|_x_j[χ^n-1]|_x_j≤ C Δ t ‖χ^n-1‖^2 + C Δ t   h^2k+1.

The fourth term in (<ref>) is bounded by Cauchy-Schwarz's inequality,  trace inequalities, approximation results, the 
CFL condition and (<ref>):

    1/2Δ t ∑_j=1^N α(u_h^n-1)|_x_j [χ^n-1-η^n-1]|_x_j[χ^n-1-χ^n]|_x_j   ≤‖χ^n-1-χ^n‖^2 + C Δ t^2 h^-2‖χ^n-1‖^2 + C Δ t^2   h^2k
       ≤‖χ^n-1-χ^n‖^2  +  C Δ t ‖χ^n-1‖^2 + C Δ t h^2k+2.

The fifth term in (<ref>) is handled exactly like the fourth term.
Similarly the first part in the sixth term has the following bound:

    Δ t ∑_j=1^N  α(u_h^n)|_x_j [χ^n-1-χ^n]|_x_j [χ^n]|_x_j≤‖χ^n-1-χ^n‖^2 + C Δ t ‖χ^n‖^2.

For the second part, we use a Taylor expansion in time and the CFL condition:

    Δ t ∑_j=1^N  α(u_h^n)|_x_j [η^n-1-η^n]|_x_j [χ^n]|_x_j≤ C Δ t^2 h^k‖χ^n‖≤ C Δ t ‖χ^n‖^2 + C Δ t   h^2k+2.

The last two terms in (<ref>) are treated almost identically, using approximation results, and
the boundedness of α:

    1/2Δ t ∑_j=1^N α(u_h^n)|_x_j [η^n]|_x_j [χ^n]|_x_j   +1/2Δ t ∑_j=1^N α(u_h^n-1)|_x_j [η^n-1]|_x_j [χ^n-1]|_x_j≤ C ε^-1Δ t   h^2k+1 
    + εΔ t ∑_j=1^N α(u_h^n)|_x_j [χ^n]^2|_x_j
       + εΔ t ∑_j=1^N α(u_h^n-1)|_x_j [χ^n-1]^2|_x_j,  ∀ε>0.

To summarize, with (<ref>), the term θ_3 is bounded as:

    θ_3 ≤   C Δ t (‖χ^n‖^2+‖χ^n-1‖^2) 
    + C Δ t^6 
    + C Δ t  (1+ε^-1) h^2k+1
       -(1/2-ε) Δ t ∑_j=1^N α(u_h^n)|_x_j [χ^n]^2|_x_j
    -(1/2-ε) Δ t ∑_j=1^N α(u_h^n-1)|_x_j [χ^n-1]^2|_x_j,   ∀ε >0,    n≥ 2.

For n=1, the term θ_3 is simply bounded as:

    θ_3 ≤ C Δ t ‖χ^1‖^2
    + C Δ t  (1+ε^-1) h^2k+1
    -(1/2-ε) Δ t ∑_j=1^N α(u_h^1)|_x_j [χ^1]^2|_x_j
    + 2‖χ^1‖^2,
    ∀ε >0.


Combining the bounds above for θ_i, 1≤ i≤ 3, we conclude the proof.





§ PROOF OF THEOREM <REF>




The proof for the forward Euler scheme is also done by induction.  It is a less technical proof than
for the Adams–Bashforth scheme.  We skip many details and give an outline of the proof.
Denote

    ξ^n = ũ_h^n-Π_h u^n.

The induction hypothesis is less restrictive than for the Adams-Bashforth method, which yields
a convergence result that is valid for polynomials of degree one and above.

    ξ^ℓ≤ h,   ∀  0≤ℓ≤ M.

Since ξ^0=0, the hypothesis (<ref>) is trivially satisfied for ℓ=0.  
Fix ℓ∈{1,…,M} and assume that 

    ξ^n≤ h,   ∀  0≤ n ≤ℓ-1.

We now have to  show that (<ref>) is valid for n = ℓ.  We begin by deriving an error inequality.
We fix an interval I_j for 0≤ j≤ N.  Using consistency in space of the scheme:

    ∫_I_j    u_t^n ϕ_h = ℋ_j(u^n, ϕ_h),    0≤ n≤ M,

we obtain, after some manipulation, the error equation:

    ∫_I_j(ξ^n+1 - ξ^n) ϕ_h = ∫_I_j(Δ t   u_t^n  - u^n+1 + u^n ) ϕ_h + ∫_I_j(η^n+1 - η^n )  ϕ_h + Δ t(ℋ_j(u_h^n, ϕ_h) - ℋ_j(u^n, ϕ_h)).

The first term in the right-hand side of (<ref>) is bounded using a Taylor expansion, whereas 
the second term vanishes due to (<ref>).  Summing over the elements from j = 0, …, N 
results in

    ∫_0^L (ξ^n+1 - ξ^n ) ϕ_h     ≤  C Δ t^2 ∫_0^L |ϕ_h|  + Δ t ∑_j = 0^N ( ℋ_j(u_h^n, ϕ_h) - Δ t  ℋ_j(u^n, ϕ_h) ).

Define

    b̃^n(ϕ_h) = Δ t ∑_j = 0^N ( ℋ_j(u_h^n, ϕ_h) - Δ t  ℋ_j(u^n, ϕ_h) ).

Then equation (<ref>) becomes

    ∫_0^L (ξ^n+1 - ξ^n ) ϕ_h     ≤  C Δ t^2 ∫_0^L |ϕ_h|  +b̃^n(ϕ_h),

and Cauchy Schwarz's and Young's inequalities imply

    ∫_0^L (ξ^n+1 - ξ^n ) ϕ_h     ≤  C Δ t^3 + C Δ t  ϕ_h^2  +b̃^n(ϕ_h).

We now choose ϕ_h = ξ^n to obtain:

    ∫_0^L (ξ^n+1 - ξ^n ) ξ^n  ≤  C Δ t^3 + C Δ t  ξ^n^2 + b̃^n(ξ^n).

It then follows that

    1/2ξ^n+1^2    - 1/2ξ^n^2 ≤1/2ξ^n+1 - ξ^n^2 +  C Δ t^3 + C Δ t  ξ^n^2 + b̃^n(ξ^n).

The terms ‖ξ^n+1-ξ^n‖  and b̃^n(ξ^n) are bounded by:

    ‖ξ^n+1-ξ^n‖^2 ≤ C Δ t^4 + C Δ t ‖ξ^n‖^2 + C Δ t  h^2k+2, 
    
    b̃^n(ξ^n) ≤ C Δ t ‖ξ^n‖^2 + C Δ t   h^2k+1.

Proof of (<ref>) follows closely the proof of Lemma <ref> but is
less technical. We skip it.  Proof of (<ref>) differs from the proof of Lemma <ref> and details are given in
Appendix <ref>.  
The error inequality simplifies to:

    ξ^n+1^2 - ξ^n^2 ≤ C Δ t^3 +  C Δ t  ξ^n^2 + C Δ t h^2k+1.

Summing from n = 0, …, ℓ-1, and using the fact that ξ^0 = 0,  one obtains:

    ξ^ℓ^2    ≤ C Δ t^2 + C h^2k+1 + C Δ t ∑_n=0^ℓ-1ξ^n^2.

We now apply Gronwall's inequality:

    ξ^ℓ^2 ≤ C_4 T e^T (Δ t^2 +  h^2k+1),

where C_4 is independent of ℓ.  Employing the CFL condition Δ t = O(h^2), one has:

    ξ^ℓ≤(C_4 T e^T )^1/2( Δ t +  h^k+1/2) 
    = (C_4 T e^T )^1/2( h^2 +  h^k+1/2).
 
Hence the induction is complete if h is small enough so that

    C_4 T e^T h < 1.

Since η^ℓ≤ C h^k+1 and u^ℓ - u_h^ℓ≤η^ℓ + ξ^ℓ one obtains:

    u^ℓ - u_h^ℓ≤ C (Δ t +  h^k+1/2),

and we conclude the proof.






§ NUMERICAL RESULTS





 §.§ Scalar case


In this section, we use the method of manufactured solutions to numerically verify convergence rates.  Solutions to the inviscid Burger's equation, 

    ∂ u/∂ t + ∂/∂ x( 1/2 u^2 ) = 0,

are approximated using the Adams–Bashforth scheme (<ref>).
We consider the following exact solution to (<ref>) posed in the interval [0,1]:

    u(x,t) = cos(2 π x) sin(t) + sin(2 π x) cos(t).

Convergence rates in space, given in Table <ref>, are calculated for polynomial degrees k = 1, 2, 3 by fixing a small timestep Δ t = 10^-4  so the temporal error is small compared to the spatial error.  The spatial discretization parameter h = 1/2^m for m = 1, …, 5, and we evolve the solution for ten timesteps.  Our results yield a rate of k+1 in space, verifying the fact that the convergence estimate in Theorem <ref> is suboptimal. 

Errors and rates in time are provided in Table <ref>.  We fix h = 1/4, vary Δ t = 1/2^m, m = 10, …, 13, and consider high polynomial degrees k = 8,9 so the spatial error is smaller than the temporal error.  We evolve the solution to the final time T = 1 s.  We recover the expected second order rate in time.




 §.§ System case


In this section we compute convergence rates for a hyperbolic system that is the motivation for this work: a model which describes one–dimensional blood flow in an elastic vessel:

    ∂/∂ t[ A; Q ]
    +
    ∂/∂ x[                    Q; αQ^2/A + 1/ρ(Aψ - Ψ) ]
    =
    [             0; -2 πνα/α-1Q/A ], 
       p = p_0 + ψ(A; A_0),   Ψ = ∫_A_0^A ψ(ξ;A_0) d ξ.

The variables are vessel cross sectional area A and fluid momentum Q.  The parameters are the reference pressure p_0 = 0 dynes/cm^2, the reference cross sectional area A_0 = 1 cm^2, the non–dimensional Coriolis coefficient α = 1.1, the fluid density ρ = 1.06 g/cm^3, and the kinematic viscosity ν = 3.302 × 10^-2 cm^2/s.  For these computations we use a typical form for the function relating area to pressure <cit.>:

    ψ = β(A^1/2 - A_0^1/2),
 
with β = 1 dynes/cm^3.  In defining the numerical flux for our computations, we use a version of the local Lax–Friedrichs flux suggested for nonlinear hyperbolic systems in <cit.>.  With U = [A,Q]^T and λ_1( U) and λ_2( U) the eigenvalues of the Jacobian of the flux function in (<ref>), the flux is defined with:

    J( U^-|_x_j,  U^+|_x_j) = max( |λ_1( U^-|_x_j)|, |λ_1( U^+|_x_j)| , |λ_2( U^-|_x_j)| , |λ_2( U^+|_x_j)|   ).


To compute errors and rates, we solve (<ref>) in the interval [0,1] with the following exact solution:

    A(x,t) = cos(2 π x) cos(t) + 2,     Q(x,t) = sin(2 π x) cos(t).

The discretization for a hyperbolic system follows the same procedure as for a scalar hyperbolic equation.  For these simulations, we employ the second–order Adams–Bashforth scheme  (<ref>) with the local Lax–Friedrichs numerical flux.

Errors and convergence rates in space, provided in Tables <ref> and  <ref> , are determined by fixing a small time step Δ t = 2 × 10^-5 s and taking h = 1 / 2^m for m = 1, … 5.  We consider k = 1, 2, 3 and evolve the solution for ten time steps.



To calculate the rate in time, we make the error in space small by choosing high order polynomials k = 8,9 on a mesh with size h=1/4.  By taking h to be constant, we avoid overly refining Δ t due to the CFL condition.  The time step Δ t = 1/2^m for m = 10, …, 13 and we evolve the solution to the final time T = 1 s.  Results are displayed in Tables <ref> and <ref>.


 
The computed rates in space and time indicate that results analogous to Theorems <ref> and <ref> can be expected for such numerical discretizations of nonlinear hyperbolic systems.  Numerical analysis for systems will be the subject of future work.



§ CONCLUSIONS


In this paper we prove a priori error estimates for fully discrete schemes approximating scalar conservation laws, where the spatial discretization is a discontinuous Galerkin method and the temporal discretization is either the second order Adams–Bashforth method or the forward Euler method.  The estimates are valid for polynomial degree greater than or equal to two for the second order method and greater than or equal to one
for the first order method in time. A CFL condition of the form Δ t = O(h^2) is required.  In future work, we will consider a priori error estimates for numerical methods approximating nonlinear hyperbolic systems like those describing blood flow in an elastic vessel.




§ APPENDIX




 §.§ Proof of bound (<ref>)



Using Taylor expansions up to third order, we write

    f(u_h^n) - f(u^n)    = f'(u^n)(u_h^n - u^n) +1/2 f”(u^n)(u_h^n-u^n)^2
    +1/6 f”'(ζ_1^n) (u_h^n-u^n)^3 
       = f'(u^n)(χ^n-η^n) +1/2 f”(u^n)(χ^n-η^n)^2 
    +1/6 f”'(ζ_1^n) (χ^n-η^n)^3 ,  
       = f'(u^n)χ^n 
    +1/2 f”(u^n) (χ^n)^2 
    -f'(u^n) η^n 
    - f”(u^n)  χ^n η^n 
    +1/2 f”(u^n) (η^n)^2 
    +1/6 f”'(ζ_1^n) (χ^n-η^n)^3
    
       = β_1+…+β_6,
    
    	f({u_h^n}) - f(u^n)    = f'(u^n)({u_h^n} - {u^n}) 
    +1/2 f”(u^n)({u_h^n}-u^n)^2
       =  f'(u^n)({χ^n} - {η^n}) + 1/2 f”(u^n)({χ^n}-{η^n})^2
    +1/6 f”'(ζ_2^n) ({χ^n}-{η^n})^3,
       = f'(u^n){χ^n} 
    +1/2 f”(u^n) ({χ^n})^2 
    -f'(u^n) {η^n} 
    - f”(u^n) {χ^n}{η^n}
       
    +1/2 f”(u^n) ({η^n})^2 
    +1/6 f”'(ζ_2^n) ({χ^n}-{η^n})^3
    
       = γ_1+…+γ_6,

where ζ^n_1 and ζ_2^n are some points between u_h^n and u^n, and {u_h^n} and u^n respectively.
We substitute these expansions in the terms ℱ(n,χ^n) and write:

    ℱ(n,χ^n)    = X_1 + … + X_6,

with

    X_i = Δ t ∑_j = 0^N  ∫_I_jβ_id χ^n/dx 
    - Δ t ∑_j = 1^N γ_i |_x_j [χ^n]|_x_j,    1≤ i≤ 6.

We integrate by parts the first term in the definition of X_1 and use the fact that f' vanishes at the endpoints of the domain, namely
at x_0 and x_N+1. The term X_1 then simplifies to

    X_1 =  -1/2Δ t ∑_j=0^N ∫_I_j (∂/∂ x f'(u^n)) (χ^n)^2
    ≤ C Δ t ‖χ^n ‖^2.

Using the assumption ‖χ^n‖≤ h^3/2 and trace inequalities, we have

    X_2     = 1/2Δ t ∑_j=0^N ∫_I_j f”(u^n) (χ^n)^2 d χ^n/dx
    -1/2Δ t ∑_j=1^N f”(u^n)|_x_j  ({χ^n})^2|_x_j [χ^n]|_x_j
       ≤ C Δ t ‖χ^n‖_∞ h^-1‖χ^n‖^2
    ≤ C Δ t  ‖χ^n‖^2.

To bound the term X_3 we define the following piecewise
constant function u_c^n elementwise as:

    u_c^n|_I_j(x) = u^n|_x_j,   ∀ x ∈ I_j,   ∀ 0≤ j ≤ N.

We  note that 

    f'(u^n) - f'(u_c^n)_∞≤ Ch.

We then rewrite the term X_3

    X_3     =  -Δ t ∑_j = 0^N ∫_I_j f'(u^n)η^n  d χ^n/dx 
    + Δ t ∑_j = 1^N  f'(u^n){η^n} |_x_j [χ^n]|_x_j
        = -Δ t ∑_j = 0^N ∫_I_j (f'(u^n)-f'(u_c^n))η^n  d χ^n/dx 
    -Δ t ∑_j = 0^N f'(u_c^n) ∫_I_jη^n  d χ^n/dx
       + Δ t ∑_j = 1^N  (f'(u^n)-f'({u_h^n}))|_x_j{η^n} |_x_j [χ^n]|_x_j
    + Δ t ∑_j = 1^N  f'({u_h^n})|_x_j{η^n} |_x_j [χ^n]|_x_j.

The second term above vanishes because of (<ref>).
The first term is bounded using approximation properties and (<ref>).

    Δ t ∑_j = 0^N ∫_I_j (f'(u^n)-f'(u_c^n))η^n  d χ^n/dx≤ C Δ t  h^2k+2 +
    CΔ t ‖χ^n‖^2.

Using a Taylor expansion, for some ζ_3^n we have

    f'(u^n)-f'({u_h^n}) = f”(ζ_3^n) {u^n-u_h^n}≤ C (‖χ^n‖_∞ + ‖η^n‖_∞).

Using the assumption ‖χ^n‖≤ h^3/2 we then have

    Δ t ∑_j = 1^N  (f'(u^n)-f'({u_h^n})){η^n} |_x_j [χ^n]|_x_j≤ C Δ t ‖χ^n‖^2 + C Δ t   h^2k+2.

For the last term in (<ref>) we employ (<ref>) to obtain:

    Δ t ∑_j = 1^N     f'({u_h^n})|_x_j{η^n} |_x_j [χ^n]|_x_j≤ CΔ t ∑_j = 1^N (α(u_h^n)|_x_j  + C |[u_h^n]|_x_j | ) |{η^n} |_x_j| | [χ^n]|_x_j| 
       = CΔ t ∑_j = 1^Nα(u_h^n)|_x_j|{η^n} |_x_j| | [χ^n]|_x_j| 
    + CΔ t ∑_j = 1^N |[u_h^n]|  |{η^n} |_x_j| | [χ^n]|_x_j| .

Using Cauchy-Schwarz's and Young's inequalities, approximation results and the assumption ‖χ^n‖≤ h^3/2, we obtain

    Δ t ∑_j = 1^N  f'({u_h^n})|_x_j{η^n} |_x_j [χ^n]|_x_j≤εΔ t ∑_j=1^N α(u_h^n)|_x_j  [χ^n]|_x_j^2
    + Cε^-1Δ t   h^2k+1
    + C Δ t ‖χ^n‖^2.

In summary we have

    X_3 ≤εΔ t ∑_j=1^N α(u_h^n)|_x_j  [χ^n]|_x_j^2
    + Cε^-1Δ t   h^2k+1
    + C Δ t ‖χ^n‖^2 + C Δ t  h^2k+1.

The bounds for X_4, X_5, and X_6 are standard applications of Cauchy Schwarz's inequality, Young's inequality, the induction hypothesis, assumption (<ref>), and inequalities (<ref>), (<ref>), and (<ref>)–(<ref>):

    X_4     = -Δ t ∑_j=0^N ∫_I_j f”(u_n) χ^n η^n d χ^n/dx
    +Δ t ∑_j=1^N f”(u_n)|_x_j{χ^n}|_x_j{η^n}|_x_j [χ^n]|_x_j
       ≤ C Δ t χ^n^2 + Δ t h^2k+1, 
    
    X_5     = 1/2Δ t ∑_j=0^N ∫_I_j f”(u_n)|_x_j  (η^n)^2 d χ^n/dx
    -1/2Δ t ∑_j=1^N f”(u_n) {η^n}^2|_x_j  [χ^n]|_x_j
       ≤ C Δ t h^2k+2 + C Δ t χ^n^2, 
    
    X_6    = 1/6Δ t ∑_j=0^N ∫_I_j f”'(ζ_1^n) (χ^n-η^n)^3 d χ^n/dx
    -1/6Δ t ∑_j=1^N f”'(ζ_2^n) ({χ^n}-{η^n})^3|_x_j [χ^n]|_x_j
       ≤ C Δ t χ^n^2 + C Δ t h^2k+1.

We can then conclude by combining all the bounds above. 



 §.§ Proof of bound (<ref>)



We rewrite, using the definition of α

    b̃^n(ξ^n) = θ_1 + θ_2 + θ_3,

with

    θ_1 = 
    Δ t ∑_j=0^N ∫_I_j (f(ũ_h^n)-f(u^n))dξ^n/dx
    - Δ t ∑_j=1^N (f({ũ_h^n})-f(u^n))|_x_j [ξ^n]|_x_j,


    θ_2 =  Δ t ∑_j=1^N ∫_I_j (s(ũ_h^n)-s(u^n))ξ^n,


    θ_3 = -Δ t ∑_j=1^N α(ũ_h^n)|_x_j [ũ_h^n]|_x_j [ξ^n]|_x_j.

We note that the bound for θ_1
follows the argument of the proof of (<ref>), where we substitute χ^n by ξ^n.
As in the previous section,  we use Taylor expansions up to third order and write the term b̃^n(ξ^n) as a sum
of six terms, X_i, 1≤ i≤ 6.  Bounds for X_i are obtained in a similar fashion, except for the term X_2
which is bounded differently because the 
the induction hypothesis for the forward Euler scheme is weaker than the hypothesis for the Adams–Bashforth scheme.
We have

    X_2    =  Δ t 1/2∑_j = 0^N  ∫_I_jf”(u^n)(ξ^n)^2d ξ^n/dx - Δ t 1/2∑_j = 1^N f”(u^n)({ξ^n})^2 |_x_j [ξ^n]|_x_j.

We rewrite the first term above. Integrating the first term by parts gives and using the assumption that f”' vanishes at the endpoints of the interval gives:

    Δ t 1/2∑_j = 0^N  ∫_I_jf”(u^n)(ξ^n)^2d ξ^n/dx   = Δ t 1/6∑_j = 0^N  ∫_I_jf”(u^n)d (ξ^n)^3/dx
       = Δ t 1/6∑_j = 1^N f”(u^n)|_x_j [(ξ^n)^3]|_x_j -Δ t 1/6∑_j = 0^N ∫_I_j∂ f”(u^n)/∂ x (ξ^n)^3.

Now, we use the identity [ξ^3] = 2{ξ}^2[ξ] + {ξ^2}[ξ] to rewrite the first term in the right-hand side of (<ref>):

    X_2    = Δ t 1/6∑_j = 1^N f”(u^n)|_x_j( {(ξ^n)^2} - {ξ^n}^2) [ξ^n]|_x_j - Δ t 1/6∑_j = 0^N ∫_I_j∂ f”(u^n)/∂ x (ξ^n)^3.

Employing the identity {ξ^2} - {ξ}^2  = 1/4[ξ]^2 for the first term and inductive hypothesis ξ^n_∞≤ h^1/2 on the second term gives:

    X_2    ≤Δ t 1/24∑_j = 1^N f”(u^n)|_x_j  [ξ^n]^3|_x_j + C Δ t ξ_∞ξ^2 ≤Δ t 1/24∑_j = 1^N f”(u^n)|_x_j  [ξ^n]^3|_x_j + C Δ t ξ^2.

The first term in (<ref>) is broken into two parts:

    Δ t 1/24∑_j = 1^N f”(u^n)|_x_j  [ξ^n]^3|_x_j   = Δ t 1/24∑_j = 1^N (f”(u^n)|_x_j - f”({ũ_h^n})|_x_j)  [ξ^n]^3|_x_j +   Δ t 1/24∑_j = 1^N f”({ũ_h^n})|_x_j [ξ^n]^3|_x_j.

We use for the first term in (<ref>) a Taylor expansion f”(u^n) - f”({ũ_h^n}) = f”'(ζ^n) {η^n-ξ^n} 
with the inductive hypothesis to obtain the following bound:

    Δ t 1/24∑_j = 1^N (f”(u^n)|_x_j-f”(u^n_h)|_x_j)  [ξ^n]^3|_x_j
            ≤ CΔ t ξ^n^2.

For the last term in (<ref>), since [u^n] = 0, we rewrite it using the identity [ξ^n] = [η^n] + [ũ_h^n]:

    Δ t 1/24∑_j = 1^N f”({ũ_h^n})|_x_j [ξ^n]^3|_x_j    
    = Δ t 1/24∑_j = 1^N f”({ũ_h^n})|_x_j [η^n] |_x_j[ξ^n]^2|_x_j 
    + Δ t 1/24∑_j = 1^N f”({ũ_h^n})|_x_j [ũ_h^n]|_x_j [ξ^n]^2|_x_j.

The first term in (<ref>) can be estimated with trace inequalities and approximation results. The second term in (<ref>) is bounded using inequality (<ref>) and the induction hypothesis:

    Δ t 1/24∑_j = 1^N f”({ũ_h^n})|_x_j [ξ^n]^3|_x_j   ≤ C Δ t h^-1η^n_∞ξ^n^2 
    + Δ t 1/3∑_j = 1^N (α(ũ_h^n)|_x_j + C |[ũ_h^n]|^2|_x_j) [ξ^n]^2|_x_j
       ≤ C Δ t ξ^n^2 + Δ t 1/3∑_j = 1^Nα(ũ_h^n)|_x_j  [ξ^n]^2|_x_j + C Δ t h^-1ũ_h^n_∞^2 ξ^n^2 
       ≤ C Δ t ξ^n^2 + Δ t 1/3∑_j = 1^Nα(ũ_h^n)|_x_j  [ξ^n]^2|_x_j.

Combining all the estimates gives:

    X_2 ≤Δ t 1/3∑_j = 1^Nα(ũ_h^n)|_x_j  [ξ^n]^2|_x_j + CΔ tξ^n^2.

This bound is added to the bounds for the other terms X_i's to obtain:

    θ_1 ≤ C Δ t ‖ξ^n‖^2 
    + (1/3 + ε) ∑_j = 1^Nα(ũ_h^n)|_x_j [ξ^n]^2|_x_j 
    + C Δ t (1+ε^-1) h^2k+1.

The term θ_2 is bounded using Lischitz continuity of s:

    θ_2 ≤ C Δ t ‖ξ^n‖^2 + C Δ t   h^2k+2.

The term θ_3 is rewritten as

    θ_3 = -Δ t ∑_j=1^N α(ũ_h^n)|_x_j [ξ^n]^2|_x_j
    + Δ t ∑_j=1^N α(ũ_h^n)|_x_j [η^n]|_x_j [ξ^n]|_x_j.

Using Young's inequality and approximation results we obtain

    θ_3 ≤ (-1+ε) Δ t ∑_j=1^N α(ũ_h^n)|_x_j [ξ^n]^2|_x_j
    + C Δ t  h^2k+1,   ∀ε>0.

This means that by choosing ε = 1/3 in the above, we conclude

    b̃^n(ξ^n) ≤ C Δ t ‖ξ^n‖^2 + C Δ t   h^2k+1.



plain


