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On monotonicity of finite-difference schemes. (English. Russian original) Zbl 0916.65093

Sib. Math. J. 39, No. 5, 959-972 (1998); translation from Sib. Mat. Zh. 39, No. 5, 1111-1126 (1998).
Conditions are studied under which an explicit linear one-dimensional scheme solution and its finite difference derivatives are free of oscillations. The equation \(u_t+u_x=0\) is considered. As the author shows, one of the above-mentioned conditions is strong monotonicity of the scheme, namely, after one time step, the scheme transforms any distribution with a single extremum to a distribution with a single extremum.
The following theorem is proven: The scheme \[ u^{n+1}_i=\sum_k c_k u^{n}_{i+k} \] is strongly monotone, if the following conditions are fulfilled: \[ c_k \geq \sqrt {c_{k+1}c_{k-1}} \quad\forall k, \]
\[ \forall l >k : c_kc_l >0 \Rightarrow c_i>0 \quad\forall i\in[k,l]. \] The above conditions are more restrictive than the usual nonnegativity condition for the coefficients. For example, the following three-point scheme family \[ u^{n+1}_i=u^{n}_{i}-\lambda(u^{n}_{i+1}-u^{n}_{i-1})/2+ C\lambda(u^{n}_{i+1}-2u^{n}_{i}+u^{n}_{i-1}), \qquad \lambda=\tau/h, \] is strongly monotone, if \(0.5\leq C\leq (2-\sqrt{1-3\lambda^2/4})/(3\lambda)\). The usual nonnegativity condition for the coefficients is equivalent to the relations \(0.5\leq C\leq 1/(2\lambda)\). A four-point scheme family is also considered. The results of test calculations are presented.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
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