THE CFL CONDITION FOR SPECTRAL APPROXIMATIONS TO HYPERBOLIC INITIAL-BOUNDARY VALUE-PROBLEMS
THE CFL CONDITION FOR SPECTRAL APPROXIMATIONS TO HYPERBOLIC INITIAL-BOUNDARY VALUE-PROBLEMS
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DOI：
10.1090/s0025-5718-1991-1066833-9
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发表时间：
1991-04-01
影响因子：
2
通讯作者：
TADMOR, E
中科院分区：
文献类型：
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作者：
GOTTLIEB, D;TADMOR, E
研究变系数标量双曲型初边值问题谱逼近的稳定性。 时间离散显式多级或龙格-库塔法的顺序小于或等于3（向前欧拉时间差分包括在内），我们研究空间离散的谱和伪谱近似与一般家庭的雅可比多项式。 我们证明了这些完全显式谱近似是稳定的，只要它们的时间步长Δ t受到CFL类条件Δ t <常数N-2的限制，其中N等于空间自由度的数量。 我们给出了两个独立的证明这一结果，根据两个不同的选择适当的L2加权范数。 在这两种方法中，证明都取决于一个有趣的逆不等式。 我们的结果证实了普遍持有的信念，即上述CFL稳定性限制，这是广泛使用的实际实现，保证了稳定性（因此收敛）的全显式谱近似在非周期的情况下。
We study the stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients. Time is discretized by explicit multi-level or Runge-Kutta methods of order less-than-or-equal-to 3 (forward Euler time-differencing is included), and we study spatial discretizations by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. We prove that these fully explicit spectral approximations are stable provided their time step, DELTA-t, is restricted by the CFL-like condition DELTA-t < Const. N-2, where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of appropriate L2-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. Our result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.