Investigation of Explicit Numerical Methods for Solving Parabolic Equations

Introduction. Numerical methods for solving heat conduction equations are widely used in various fields of science and technology. The development of new improved numerical algorithms for solving parabolic equations makes it possible to increase the accuracy of calculations, remove restrictions on the integration step in time, which will allow us to obtain solutions to practical problems in an acceptable time with high accuracy. The purpose of this work is to study the explicit methods for solving parabolic type equations using the example of solving the equation of thermal conductivity with variable coefficients. Materials and Methods. The paper provides a software implementation of numerical algorithms for solving parabolic type equations: the well-known explicit scheme, the local iteration method, and the hyperbolization method. The algorithms were implemented in C++. The order of approximation of algorithms and their execution time for solving an equation with an analytical solution are analyzed. Results. A numerical study of the schemes has shown that each of them has a second order of approximation in space. However, the estimated time to solve the problem using the hyperbolization method is less. Discussion and Conclusion. In the case of solving systems of equations of the parabolic type, if it is necessary to use explicit schemes, the hyperbolization method will significantly reduce the estimated time of the problem while maintaining the stated order of accuracy.