The paper presents an enhanced analysis of the Lax‐Wendroff difference scheme—up to the eighth‐order with respect to time and space derivatives—of the modified‐partial differential equation (MDE) of the constant‐wind‐speed advection equation. The modified equation has been so far derived mainly as a fourth‐order equation. The Π‐form of the first differential approximation (differential approximation or equivalent equation) derived by expressing the time derivatives in terms of the space derivatives is used for presenting the MDE. The obtained coefficients at higher order derivatives are analyzed for indications of the character of the dissipative and dispersive errors. The authors included a part of the stencil applied for determining the modified differential equation up to the eighth‐order of the analyzed modified differential equation for the second‐order Lax‐Wendroff scheme. Neither the derived coefficients at the space derivatives of order p ∈ (7 – 8) in the modified differential equation for the Lax‐Wendroff difference scheme nor the results of analyses on the basis of these coefficients of the group velocity, phase shift errors, or dispersive and dissipative features of the scheme have been published. The MDEs for 2 two‐step variants of the Lax‐Wendroff type difference schemes and the MacCormack predictor–corrector scheme (see MacCormack's study) constructed for the scalar hyperbolic conservation laws are also presented in this paper. The analysis of the inviscid Burgers equation solution with the initial condition in a form of a shock wave has been discussed on their basis. The inviscid Burgers equation with the source is also presented. The theory of MDE started to develop after the paper of C. W. Hirt was published in 1968.

The relationship between radar reflectivity (Z) and rainfall intensity (R) plays a crucial role in estimating precipitation and serves as a foundation for flood risk assessment. However, empirical Z–R relationships often introduce considerable uncertainty, making the correction of rainfall estimation errors a key challenge in remote-sensing-based applications. Developing an effective approach to reduce these deviations is, therefore, essential to improve the accuracy of radar-based precipitation measurements. This study aims to develop a methodology for analyzing radar-derived precipitation using dual-polarization radar measurements, with validation based on rain gauge observations. Three well-established Z–R relationships—Marshall–Palmer, Muchnik, and Joss—were applied to radar reflectivity values measured at two heights, 1 km and 1.5 km above ground level. The Marshall–Palmer relationship applied at a height of 1.5 km yielded the smallest deviations from rain gauge measurements. Both the mean absolute error (MAE) and average precipitation difference at this height were consistent, amounting to 1.99 mm, compared to 2.32 mm at 1 km. The range of deviations in all cases was 0.54–7.64 mm at 1.5 km and 0.65–7.18 mm at 1 km. Furthermore, all tested Z–R relationships demonstrated a strong linear correlation with rain gauge data, as indicated by a Pearson correlation coefficient of 0.98. These findings enable the identification of the most accurate Z–R relationships and optimal measurement heights for radar-based precipitation estimation. These results may have important implications for operational applications and the calibration of radar precipitation products.

The Laplace operator is a differential operator which is used to detect edges of objects in digital images. This paper presents the properties of the most commonly used third-order 3x3 pixels Laplace contour filters including the difference schemes used to derive them. The authors focused on the mathematical properties of the Laplace filters. The basic reasons of the differences of the properties were studied and indicated using their transfer functions and modified differential equations. The relations between the transfer function for the differential Laplace operator and its difference operators were described and presented graphically. The impact of the corner elements of the masks on the results was discussed. This is a theoretical work. The basic research conducted here refers to a few practical examples which are illustrations of the derived conclusions.We are aware that unambiguous and even categorical final statements as well as indication of areas of the results application always require numerous experiments and frequent dissemination of the results. Therefore, we present only a concise procedure of determination of the mathematical properties of the Laplace contour filters matrices. In the next paper we shall present the spectral characteristic of the fifth order filters of the Laplace type.

The analysis of the modified partial differential equation (MDE) of the constant wind speed advection equation explicit difference scheme up to the eighth order with respect to both space and time derivatives is presented. So far, in majority of publications this modified equation has been derived mainly as a fourth-order equation. The MDE is presented in the so-called Π-form of the first differential approximation. This form includes only the space derivatives of higher order p and their coefficients μ(p). Analysis of these coefficients provides indications of the nature of the dissipative and dispersive errors. A fragment of the stencil for determining the modified differential equation up to the eighth-order MDE for the second-order Beam–Warming scheme is included. The derived coefficients μ(p) as well as the analysis of the phase shift errors, the phase and group velocities and dispersive features on the basis of these coefficients have not been published so far. The dissipative features of this method we present in [33].

This paper is a continuation of [38]. The analysis of the modified partial differential equation (MDE) of the constant-wind-speed linear advection equation explicit difference scheme up to the eighth-order derivatives is presented. In this paper the authors focus on the dissipative features of the Beam–Warming scheme. The modified partial differential equation is presented in the so-calledΠ-form of the first differential approximation. The most important part of this form includes the coefficientsμ(p) at the space derivatives. Analysis of these coefficients provides indications of the nature of the dissipative errors. A fragment of the stencil for determining the modified differential equation for the Beam–Warming scheme is included. The derived and presented coefficientsμ(p) as well as the analysis of the dissipative features of this scheme on the basis of these coefficients have not been published so far.

This study uses dual-polarization radar data from two weather radars to assess the accuracy of estimating rainfall rate ( R ) from radar reflectivity ( Z ) using classical empirical Z–R relationships (Marshall–Palmer, Wexler, and Doumoulin–Cogombles). Three additional polarimetric relationships incorporating differential reflectivity (ZDR) were also analyzed. Calculations were carried out for seven altitude levels, representing a novel approach to verifying the vertical structure of the precipitation field. Radar data were validated against measurements from a rain gauge network, enabling an assessment of spatiotemporal consistency and rainfall estimation errors. The analysis covers measurements collected during the flood event that occurred in Poland in September 2024 (Genoa low-pressure system). The results indicate that for the analyzed extreme flood event, the most reliable precipitation totals were obtained using the polarimetric relationship based on ZDR (RMSE=2.20 mm±0.90 mm, MAE=1.84 mm±0.73 mm, and Bias=-0.67 mm±0.81 mm). It was demonstrated that under the studied extreme conditions, the proposed ZDR3 relationship exhibits approximately 69% lower Bias compared with the standard operational Marshall-Palmer method. The findings confirm the potential of polarimetric methods for rainfall estimation and their applicability in operational settings during similar extreme weather events, particularly in early warning and crisis management systems.