Blow-up of finite-difference solutions to nonlinear wave equations
Blow-up of finite-difference solutions to nonlinear wave equations
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发表时间：
2016-01
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通讯作者：
Norikazu Saito;Takiko Sasaki
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文献类型：
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作者：
Norikazu Saito;Takiko Sasaki
给出了计算一维非线性波动方程爆破解的有限差分格式。通过应用时间增量控制技术，我们可以引入一个数值爆破时间，它是非线性波动方程精确爆破时间的近似。在验证了所提格式的收敛后，我们证明了有限差分格式的解在数值爆破时间内实际上是爆破的。然后，我们证明了当离散化参数趋于零时，数值爆破时间收敛到准确的爆破时间。文中还给出了几个数值算例，验证了理论结果的正确性。
Finite-difference schemes for computing blow-up solutions of one dimensional nonlinear wave equations are presented. By applying time increments control technique, we can introduce a numerical blow-up time which is an approximation of the exact blow-up time of the nonlinear wave equation. After having verified the convergence of our proposed schemes, we prove that the solution of a finite-difference scheme actually blows up in the numerical blow-up time. Then, we prove that the numerical blow-up time convergences to the exact blow-up time as the discretization parameters tend to zero. Several numerical examples that confirm the validity of our theoretical results are also offered.