The book: “Sequential Software for the Robust Multigrid Technique” by S. I. Martynenko

The repository contains examples and programs from the book “Sequential Software for the Robust Multigrid Technique” by S. I. Martynenko, Triumph, Moscow, 2020. Complete text of the book in Russian is available via the link. Abstract in English, theoretical background and description of the source code files are provided below.

Abstract of the book

This book covers sequential software for the Robust Multigrid Technique (RMT) and presents basic concepts of modern numerical methods for mathematical modeling of physical and chemical processes in the computational continuum mechanics (thermal conductivity, chemical hydrodynamics, convective heat transfer, electrodynamics, etc.). FORTRAN language subroutines for solving the boundary value problems of the computational continuum mechanics by point and block Seidel method (Vanka-type smoother) are given in Chapters 1-4. The multigrid module RMT_3D_2020 is described in detail in Chapter 2 of this book. The module contains subroutines for implementation of the problem-independent components of the Robust Multigrid Technique for solving the three-dimensional boundary value problems. Examples of the multigrid module RMT_3D_2020 application in static and dynamic cycles are given in Chapters 3 and 4. Chapter “Short history of RMT” represents the historical development of the Robust Multigrid Technique.

Potential readers are graduate students and researchers working in applied and numerical mathematics as well as multigrid practitioners and software programmers for modeling physical and chemical processes in aviation and space industries, power engineering, chemical technology and other branches of mechanical engineering.

Mathematical background of the Robust Multigrid Technique

All the required mathematical background is provided in the book: S. I. Martynenko “The Robust Multigrid Technique: For Black-Box Software”, de Gruyter, Berlin, 2017 (https://www.degruyter.com/view/title/527481).

This book presents a detailed description of a robust pseudomultigrid algorithm for solving (initial-)boundary value problems on structured grids in a black-box manner. To overcome the problem of robustness, the presented Robust Multigrid Technique (RMT) is based on the application of the essential multigrid principle in a single grid algorithm. It results in an extremely simple, very robust and highly parallel solver with close-to-optimal algorithmic complexity and the least number of problem-dependent components. Topics covered include an introduction to the mathematical principles of multigrid methods, a detailed description of RMT, results of convergence analysis and complexity, possible expansion on unstructured grids, numerical experiments and a brief description of multigrid software, parallel RMT and estimations of speed-up and efficiency of the parallel multigrid algorithms, and finally applications of RMT for the numerical solution of the incompressible Navier-Stokes equations.

List of modules

• Multigrid module RMT_3D_2020 (§ 1, § 3 chapter 2) includes subprograms for main components of RMT for solving 3D boundary value problems. The multigrid module is available as a static library only for Intel Fortran Compiler under Windows and GFortran under Linux operation systems.

• Module RMT_3D_UFG (§ 2 chapter II, p. 70) includes the subprogram U_Finest_Grid for the uniform grid generation using (2.1)–(2.4).

• Module RMT_3D_NFG (§ 6 chapter II, p. 93) includes subprograms Finest_Grid and CELLS for the nonuniform grid generation. Finite volume faces are located between the vertices.

• Module RMT_3D_NFG2 (§ 6 chapter II, p. 97) includes subprograms Finest_Grid and CELLS for the nonuniform grid generation. Vertices are located between the finite volume faces.

• Module RMT_3D_TIMES includes a basic time measurement function.

List of example programs

1. Program RMT_3D_2020_Example_01.f90 (§ 2 chapter I, p. 45): solution of discrete Poisson equation (1.6) with the boundary conditions (1.7) by point Seidel method.

2. Program RMT_3D_2020_Example_02_M.f90 (§ 3 chapter I, p. 54): solution of discrete Poisson equation (1.6) with the boundary conditions (1.7) by block Seidel method. Used modules: RMT_3D_2020_Example_02_1, RMT_3D_2020_Example_02_2 (p. 60), RMT_3D_TIMES.

3. Program RMT_3D_2020_Example_03.f90 (§ 2 chapter II, p. 75): formation of a multigrid structure generated by a uniform grid without information files.

4. Program RMT_3D_2020_Example_04.f90 (§ 4 chapter II, p. 79): formation of a multigrid structure generated by a uniform grid with information files.

5. Program RMT_3D_2020_Example_05.f90 (§ 5 chapter II, p. 82): example of calling the subroutine Index_Mapping for computing the index mapping.

6. Program RMT_3D_2020_Example_06.f90 (§ 5 chapter II, p. 88): approximation of derivatives on a multigrid structure. Subroutine CELLS (§ 6 chapter II, p. 92): generation a non-uniform grids with the same grid aspect ratio.

7. Program RMT_3D_2020_Example_07.f90 (§ 6 chapter II, p. 95): formation of a multigrid structure generated by a non-uniform grids with the same grid aspect ratio. Used modules: RMT_3D_NFG.

8. Program RMT_3D_2020_Example_08.f90 (§ 6 chapter II, p. 99): formation of a multigrid structure generated by a non-uniform grids with the same grid aspect ratio. Used modules: RMT_3D_NFG2.

9. Program RMT_3D_2020_Example_09_M.f90 (§ 2 chapter III, p. 107): approximation of integral (3.4) on the multigrid structure generated by a uniform grid. Used modules: RMT_3D_2020_Example_09_1.f90 and RMT_3D_2020_Example_09_2.f90.

10. Program RMT_3D_2020_Example_10_M.f90 (§ 2 chapter III, p. 111): approximation of integral (3.4) on the multigrid structure generated by a non-uniform grid. Used modules: RMT_3D_2020_Example_09_1, RMT_3D_2020_Example_09_2, RMT_3D_2020, RMT_3D_NFG.

11. Program RMT_3D_2020_Example_11_M.f90 (§ 2 chapter III, p. 114): approximation of integral (3.4) on the multigrid structure generated by a non-uniform grid. Used modules: RMT_3D_2020_Example_09_1, RMT_3D_2020_Example_09_2, RMT_3D_2020, RMT_3D_NFG2.

12. Program RMT_3D_2020_Example_12_M.f90 (§ 4 chapter III, p. 122): numerical solution of the Dirichlet problem (3.9) on a multigrid structure generated by a uniform grid. Smoother is point Seidel method. Used modules: RMT_3D_2020_Example_12_1, RMT_3D_2020_Example_12_2, RMT_3D_2020, RMT_3D_NFG2, RMT_3D_TIMES.

13. Program RMT_3D_2020_Example_13_M.f90 (§ 4 chapter III, p. 140): numerical solution of the Dirichlet problem (3.9) on a multigrid structure generated by a uniform grid. Smoother is block Seidel method. Used modules: RMT_3D_2020_Example_13_1, RMT_3D_2020_Example_13_2, RMT_3D_2020, RMT_3D_UFG, RMT_3D_TIMES.

14. Program RMT_3D_2020_Example_14.f90 (§ 4 chapter II, p. 150): index mapping in the dynamic cycle. Used modules: RMT_3D_UFG.

15. Program RMT_3D_2020_Example_15_M.f90 (§ 3 chapter IV, p. 153): approximation of integrals on the multigrid structure generated by the dynamic grids of the first levels. Used modules: RMT_3D_2020_Example_15_1, RMT_3D_2020_Example_15_2, RMT_3D_2020, RMT_3D_UFG.

16. Program RMT_3D_2020_Example_16_M.f90 (§ 4 chapter IV, p. 153): first multigrid iteration of the dynamic cycle. Used modules: RMT_3D_2020_Example_16_1, RMT_3D_2020_Example_16_2, RMT_3D_2020, RMT_3D_UFG, RMT_3D_TIMES.