## SAMG

The software package SAMG was developed for highly efficient, numerical solutions of large, sparsely populated matrix problems. Problems of this type can be found in the fields of fluid mechanics, structural mechanics, simulations of oil reservoirs and ground water, process and device simulation, semiconductor physics, and circuit simulation.

SAMG-MODFLOW for the acceleration of groundwater simulations is also available.

## Numerical Solution of Large Matrix Problems

SAMG (Algebraic Multigrid Methods for Systems) is a library of subroutines for the highly efficient solution of large linear systems of equations with sparse matrices. Such systems of equations form the numerical kernel of most simulation software packages. Usually, the numerical solution of these linear systems of equations needs most of the computational time of the whole simulation.

Compared to classical methods (e.g., the ILU-preconditioned conjugate gradient method), SAMG has the advantage of being almost unconditionally numerically scalable. This means that the computational cost using SAMG depends only linearly on the number of unknowns. Depending on the application and problem size, the computational cost can be reduced by one to two orders of magnitude. SAMG can be incorporated into an existing software package as easily as any classical method.

SAMG is available in the following versions:

- SAMG, OpenMP parallel - best suited for today's multicore computers
- SAMGp, MPI parallel - SAMG for computer clusters

## Target Group

Our work is dedicated to partners and customers involved in software development as well as applications. In addition to our solver technology, we offer analysis and advice on application problems as well as tailoring our software to the customers computer systems, especially parallel computers.

## Applications

Algebraic multigrid methods are a generalization of Geometric MultiGrid methods (GMG) which are used to solve discretized elliptic differential equations. In contrast to GMG, algebraic multigrid methods can directly be applied to the linear system without the need of geometrical background information. Hence, algebraic multigrid methods are perfectly suited to solve Partial Differential Equations (PDEs) on unstructured two- and three-dimensional grids as well as linear systems with similar properties.

Areas of Application:

- fluid mechanics
- structural mechanics
- foundry technology
- oil reservoir simulation
- groundwater simulation
- hydrothermal ore agglomeration simulation
- process simulation in semiconductor physics
- device simulation in semiconductor physics
- circuit simulation

## Background

In many applications of numerical simulation, for example in fluid flow and structural mechanics, structures and geometries are discretised by means of complex grids (see Figure). The finer the resolution such a grid has, the higher is the quality of the simulation. On the other hand, this also increases the size of the system of discretized equations which needs to be solved numerically. Due to the accuracy of simulation results required nowadays, the amount of time needed to solve these systems of equations has become a critical factor. Classical numerical solvers typically do not have the capability to solve such large systems of equations in an acceptable amount of time.

The solver module of SAMG is based on modern hierarchical techniques (Algebraic MultiGrid methods, AMG): Instead of working with the given (extremely large) system of equations, algebraic multigrid combines numerical information from a scale of increasingly coarse systems of equations in order to solve the given problem more rapidly. This coarsening process takes place automatically and is transparent for the SAMG user.

### Parallel Version of SAMG

We also offer a parallel version of SAMG (based on MPI), which can be applied to each partition of the numerical grid. As long as the number of grid cells per process is large enough, SAMG delivers exceptional performance.