Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems
Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems
复制标题
一般非线性双曲系统Glimm格式的锐收敛率
DOI：
10.1007/s00220-010-1178-5
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发表时间：
2011
影响因子：
2.4
通讯作者：
H. Jenssen
中科院分区：
文献类型：
--
作者：
A. Bressan;H. Jenssen
在一维空间中考虑一般严格双曲拟线性系统
$$u_t+A(U)u_x=0，\quad\quad(1)$$其中$${u\mapsto A(U)，u\in\omga\子集{\mathbb{R}^N}$$，是光滑矩阵值映射。给出一个总变分较小的初始数据u(0，·)，设u(t，·)是(1)的相应(唯一)消失粘性解，作为粘性抛物线逼近的解的极限uT+A(U)UX=μuxx，如μ→0.对于每一个T，证明了先验有界u^u^{varepsilon}(T，\cdot)\right\|_{{\mathbb{L}^1}}=o(1)\cdot\sqrt{\varepsilon}\，|\log{\varepsilon}|\quad\quad\quad\quad\quad\quad(1)$$for)-u(T，≥)-u(T，Glimm格式构造的近似解$$u^{varepsilon}}$$，网格尺寸$${\deltax=\Delta t={\varepsilon}}$$，并且具有采样序列的适当选择。这一结果为一般的双曲型系统提供了对满足经典Lax或Liu假设的双曲型守恒律方程组的Glimm近似解的相同类型的误差估计，该条件是关于特征值λk(U)和雅可比矩阵A(U)的特征向量rk(U)的。估计(2)是通过引入新的带三次项的波相互作用泛函得到的，该波相互作用泛函控制同一族波的非线性耦合，同时在相互作用时波强度与波速变化的乘积的量是同阶的。这正是在Glimm格式的波浪跟踪分析中产生的误差类型，这对于控制以实现收敛速度的准确估计是至关重要的。
AbstractConsider a general strictly hyperbolic, quasilinear system, in one space dimesion
$$u_t+A(u) u_x=0,\quad\quad\quad\quad\quad\quad(1)$$where $${u \mapsto A(u), u\in\Omega\subset{\mathbb{R}}^N}$$ , is a smooth matrix-valued map. Given an initial datum u(0, ·) with small total variation, let u(t, ·) be the corresponding (unique) vanishing viscosity solution of (1) obtained as a limit of solutions to the viscous parabolic approximation ut + A(u)ux = μuxx, as μ → 0. For every T ≥ 0, we prove the a-priori bound$$\left\|u^{\varepsilon}(T,\cdot)-u(T,\cdot)\right\|_{{\mathbb{L}^1}}=o(1)\cdot\sqrt{\varepsilon}\,|\log{\varepsilon}|\quad\quad\quad\quad\quad\quad(1)$$for an approximate solution $${u^{\varepsilon}}$$ of (1) constructed by the Glimm scheme, with mesh size $${\Delta x = \Delta t = {\varepsilon}}$$ , and with a suitable choice of the sampling sequence. This result provides for general hyperbolic systems the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems of conservation laws ut + F(u)x = 0 satisfying the classical Lax or Liu assumptions on the eigenvalues λk(u) and on the eigenvectors rk(u) of the Jacobian matrix A(u) = DF(u).The estimate (2) is obtained introducing a new wave interaction functional with a cubic term that controls the nonlinear coupling of waves of the same family and at the same time decreases at interactions by a quantity that is of the same order of the product of the wave strength times the change in the wave speeds. This is precisely the type of errors arising in a wave tracing analysis of the Glimm scheme, which is crucial to control in order to achieve an accurate estimate of the convergence rate as (2).