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Mathematical models of various natural processes are described by differential equations, systems of partial differential equations and integral equations. In most cases, the exact solution to such problems cannot be determined; therefore, one has to use grid methods to calculate an approximate solution using high-performance computing systems. These methods include the finite element method, the finite difference method, the finite volume method and combined methods. In this Special Issue, we bring to your attention works on theoretical studies of grid methods for approximation, stability and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. Of particular interest are new methods for solving boundary value problems with singularities, the complex geometry of the domain boundary and nonlinear equations. A part of the articles is devoted to the analysis of numerical methods developed for calculating mathematical models in various fields of applied science and engineering applications. As a rule, the ideas of symmetry are present in the design schemes and make the process harmonious and efficient.

Information technology industries --- high-order methods --- Brinkman penalization --- discontinuous Galerkin methods --- embedded geometry --- high-order boundary --- IMEX Runge–Kutta methods --- boundary value problems with degeneration of the solution on entire boundary of the domain --- the method of finite elements --- special graded mesh --- multigrid methods --- Hermitian/skew-Hermitian splitting method --- skew-Hermitian triangular splitting method --- strongly non-Hermitian matrix --- lie symmetries --- invariantized difference scheme --- numerical solutions --- finite integration method --- shifted Chebyshev polynomial --- direct and inverse problems --- Volterra integro-differential equation --- Tikhonov regularization method --- quartic spline --- triangulation --- scattered data --- continuity --- surface reconstruction --- positivity-preserving --- interpolation --- jaw crusher --- symmetrical laser cladding path --- FEPG --- wear --- high-order methods --- Brinkman penalization --- discontinuous Galerkin methods --- embedded geometry --- high-order boundary --- IMEX Runge–Kutta methods --- boundary value problems with degeneration of the solution on entire boundary of the domain --- the method of finite elements --- special graded mesh --- multigrid methods --- Hermitian/skew-Hermitian splitting method --- skew-Hermitian triangular splitting method --- strongly non-Hermitian matrix --- lie symmetries --- invariantized difference scheme --- numerical solutions --- finite integration method --- shifted Chebyshev polynomial --- direct and inverse problems --- Volterra integro-differential equation --- Tikhonov regularization method --- quartic spline --- triangulation --- scattered data --- continuity --- surface reconstruction --- positivity-preserving --- interpolation --- jaw crusher --- symmetrical laser cladding path --- FEPG --- wear