Global existence for a semi-discrete scheme of some quasilinear hyperbolic balance laws

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Abstract

In this article, we show the global existence of solution for some semi-discrete finite difference schemes of some model system of quasilinear hyperbolic balance laws with the Cattaneo law. In [10] and [11] the first author proposed the energy method for the structure-preserving fully discrete numerical schemes for semilinear partial differential equations. The result here gives its application to quasilinear partial differential equations.

Introduction

When we study partial differential equations (PDEs), nonlinear difference equations corresponding to these equations often play important roles for not only numerical simulations but also the analysis for the original problem (see e.g. [8]). Our interest directs to how to translate the methods for PDEs into corresponding difference equations. In this article we introduce the transplantation of the classical energy method for the quasilinear PDEs. As the example we consider the initial value problem for the one-dimensional quasilinear hyperbolic system:tu+xf(u)+xq=0,νtq+q+xu=0,(x,t)X×R+,u(0,x)=u0(x),q(0,x)=q0(x),xX, where ν(0,1] is a constant and u0, q0 are given initial data. We assume that f is sufficiently smooth and X=R or T. The system corresponds to various model systems describing thermoelastic material and compressible viscous fluid etc. For more precise explanation and background of this system, we refer to e.g. [2], [5] and [6]. The existence of solution to the original problem (1.1)–(1.3) is shown by the standard theory called the energy method (see e.g. [7]). More precisely, we can show the existence and uniqueness of global-in-time solution to the problem (1.1)–(1.3) under smallness assumption for initial data, which is called a small data global existence.

Let us discretize the time interval with a split size Δt>0. Denote the solutions for the discretized problem corresponding to the solution (u,q) for (1.1)–(1.2) at t=nΔt by (U(n),Q(n))=(U(n)(x),Q(n)(x)). In this article we consider the following semi-discrete mid-point rule:δn+U(n)+xf(U(n+1)+U(n)2)+x(Q(n+1)+Q(n)2)=0,νδn+Q(n)+Q(n+1)+Q(n)2+x(U(n+1)+U(n)2)=0,xX,n=0,1,2,,U(0)=u0,Q(0)=q0,xX, where δn+ is the forward finite difference operator defined by δn+U(n):=U(n+1)U(n)Δt. The solution for (1.1)–(1.2) satisfies the following lawt(12uL22+ν2qL22)+qL22=0, which is obtained by calculating ∫(1.1) × udx + ∫(1.2) × qdx. Correspondingly, the solution for (1.4)–(1.5) satisfies the law similar to (1.7) in the sense:δn+(12U(n)L22+ν2Q(n)L22)+Q(n+1)+Q(n)2L22=0, which is obtained by calculating (1.4)×U(n+1)+U(n)2dx+(1.5)×Q(n+1)+Q(n)2dx. This implies that the mid-point rule (1.4)–(1.6) inherits the energy conservation law (1.7) in the sense of (1.8). Then we also call the mid-point rule (1.4)–(1.6) a structure-preserving scheme. A numerical stability of the scheme is one of important properties. Structure-preserving discrete schemes assure the numerical stability automatically in many cases, and hence it is useful to demonstrate numerical simulations (see e.g. [1] and [9] etc.). Our claim of this article is that the procedure of existence proof for the continuous problem (1.1)–(1.3) can be carried out to the discrete scheme (1.4)–(1.6) in a similar manner. Although our method introduced here is applicable to other various models, we restrict ourselves to the simple system (1.1)–(1.2) for simplicity.

In [10] and [11], the first author proposed the energy method for the structure-preserving fully discrete schemes for semilinear PDEs. However, there the application to quasilinear PDEs is not mentioned. In general, it seems to be less results which deal with quasilinear PDEs than semilinear PDEs. Just recently, Kovács and Lubich [4] establish the interesting result corresponding to the translation of Kato's method for quasi-linear PDEs into numerical context. They consider the semi-discrete scheme as well, and show existence of local-in-time solution and its error estimate. We remark that we can show an error estimate if we consider the problem in bounded time interval following the manner introduced in [11] under some additional assumptions. However, to concentrate on our aim of transplantation and for simplicity, we restrict ourselves to show small data global existence for (1.4)–(1.6) here.

Section snippets

Main results

We shall show the small data global existence of solution for the system (1.4)–(1.6), which means the existence of {U(n),Q(n)} satisfying (1.4)–(1.6) for n=0,1,2, under the smallness assumption for initial data. Using the method explained in e.g. [7] (see also [2], [3]), we can show the global existence of the original problem (1.1)–(1.3) under the smallness assumption for initial data. In general, for the quasilinear PDEs for such as compressible fluid, thermoelasticity and so on, it is not

Local existence

We show the existence of quasilinear system (2.5) by an iteration argument. For the purpose we first consider the following linearized problem of (2.5):UU(n)Δt+f(U˜+U(n)2)x(U+U(n)2)+x(Q+Q(n)2)=0,νQQ(n)Δt+Q+Q(n)2+x(U+U(n)2)=0. Here, (U(n),Q(n))H2×H2 and (U˜,Q˜)H2×H2 are given, and (U,Q) is unknown. If there exists a solution (U,Q)H2×H2, we can define the mapping Φ by (U˜,Q˜)(U,Q) in H2×H2. The fixed point of the mapping Φ satisfying (U,Q)=Φ(U,Q) is just the solution for (2.5), and we

A priori estimate

In this section let us show the global a priori estimate for the solution. We first prove the following energy estimate.

Lemma 4.1

Let E(N) and D(N) be defined in (2.1) and (2.2). Then it holds thatE(N)+D(N)C2E(0)+P1(E(N))E(N)D(N), where C2 is independent of N.

Proof

The proof consists of a combination between energy estimates and dissipative estimates.

(1) Energy estimates. From the structure-preserving property (1.8) we have already knownE0(U(n+1),Q(n+1))E0(U(n),Q(n))+ΔtQ(n+1)+Q(n)2L22=0. In the case of s=

Acknowledgments

This work was partially supported by JSPS KAKENHI Grant Numbers JP16K05234, JP20K03687 and JP18H01131 and Grant for Basic Science Research Projects from The Sumitomo Foundation Grant Number 180823. Authors wish to express deep gratitude to prof. Norikazu Saito for his fruitful advices and the anonymous referees for their kind advices and profound comments.

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