On strong monotonicity of three-point difference schemes
Two-layer by time, three-point schemes of the form \[ \begin{gathered} u_i^{j+1} = c_{-1} u_{i-1}^j + c_{0} u_{i}^j + c_{1} u_{i+1}^j, \\ c_{-1} = (c+0.5) \lambda, \quad c_{0} =1 - 2c \lambda, \quad c_{1} = (c-0.5) \lambda, \end{gathered} \] are considered. A criterion for strong monotonicity of order \(n\) for an arbitrary \(n \in {\mathbb N}\) is obtained. Definition 1. A grid function \(u_i \) on a segment \([k,l] \), \(k,l \in {\mathbb N}\), has a local extremum whenever \( (u_k - u_{k-1}) (u_{l+1} - u_{l}) < 0\) and \(u_i = u_{i+1}\) for all \(i \in [k,l-1] \). Definition 2. A scheme \(u_i^{j+1} = \sum_k c_k u_{i+k}^j \) possesses monotonicity of order \(n+1\) whenever for every \(m \in [0,n] \), each function \(u_i \) having \(m\) local extrema is transformed by the scheme into a function \(u_i^1 \) that has at most \(m\) local extrema. Criterion. The above scheme possesses monotonicity of order \(n+1\) if and only if its coefficients satisfy the conditions \(c_{-1} \leq 0\), \(c_0 \leq \alpha_n\sqrt{ c_{-1} c_1 }\), \(c_1 \leq 0 \), where \(\alpha_n = \overline{x}_{n1} \) is the first component of the solution \((\overline{x}_{n1},\dots , \overline{x}_{nm})\), \(\overline{x}_{ni} > 0\) for all \(i\), to the following algebraic system of order \(n\): \[ \overline{x}_{n1} = \overline{x}_{ni}^{ -1} + \overline{x}_{n(i+1)} = \overline{x}_{nn}^{ -1}, \quad i = 1,2,\ldots,n-1. \] The transfer equation \(u_t + u_x = 0\) is considered for demonstrating an application of the criterion.
scientific article; zbMATH DE number 1285618