具有零阶退化方程的二阶双曲型方程奇异摄动问题的一致差分格式
Uniform Difference Scheme for a Singularly Perturbed Linear 2nd Order Hyperbolic Problem with Zeroth Order Reduced Equation
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摘要: 本文讨论了一个二阶双曲型奇异摄动问题,它的一阶导数项含有小参数ε.首先给出该问题解的能量估计及渐近解的余项估计,然后在均匀网格上构造了一个指数型拟合差分格式,最后证明了差分解在离散的能量范数意义下一致收敛于问题的精确解.Abstract: In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.
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[1] Butuzov,V.F.,Corner boundary layer in mixed singularly perturbed problem for hyperbolic equations,Matematiceskii Sbornik,104,3(1977). [2] Su Yu-cheng and Lin Ping,Numerical solution of singular perturbation problem for hyperbolic differential equation with characteristic boundaries,Proceedings of the BAIL V Conference,Shanghai,June(1988). [3] Lees,M.,Energy inequalities for the solution of differential equations,Trans.Amer.Math.Soc.,94,1(1960),58-73. [4] Doolan,E.P.,J.J.H.Miller and W.H.A.Schilders,Uniform Numerical Methods for Problems with Initial and Boundary Layers,Boole Press,Dublin(1980). -
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