It transfers solution candidates, the iterates, between different linear systems corresponding to coarser and finer meshes. The GMG method acts on the linear equation system for different mesh levels, from fine to coarse meshes. Multigrid methods provide an optimal technology for modifying, or preconditioning, the equation system for iterative techniques like the GMRES method. We can modify the linear equation system so that the GMRES method performs much better. Thanks to the AMG solver, the equations can be solved on a desktop computer.įor fluid flow problems, COMSOL Multiphysics uses the generalized minimal residual (GMRES) method, which is an iterative method for solving very large systems of linear equations.
However, we can solve the linear equations with an iterative solver using much less memory.Įven this relatively simple model of the flow in a centrifugal pump requires 350,000 equations and unknowns. The linear equation system that has to be solved in each Newton iteration is too expensive to solve with a direct solver. In our case, we have hundreds of thousands or millions of equations and unknowns, proportional to the number of nodes in the finite element mesh that we used to generate the numerical equations.
Why Should I Use Iterative Methods for Linear Equation Systems? This method is based on linearization of the nonlinear equations and solving the linear equations in a sequence of iterations, often referred to as Newton iterations, until we obtain the desired accuracy. The method for solving the systems of nonlinear equations, both for the time-dependent and stationary problems, is a damped Newton method, when the system is solved fully coupled. For steady flows, the numerical model equations form a system of nonlinear equations that has to be solved once. For transient problems, a system of nonlinear equations has to be solved at every time step. The fluid flow equations are nonlinear, which means that the discretized numerical model equations are also nonlinear. The “real” description of the geometry, to the left, is approximated with a discretized description where momentum balances and mass balances in each element are carried out.įor problems in space and time, COMSOL Multiphysics uses the method of lines, where the discretization in space is done using the finite element method and time discretization is done using some standard method for ordinary differential equations, such as backwards differentiation formula (BDF) or Generalized-α. We can say that we approximate our mathematical model with a numerical model. Instead, we can discretize in space and time in order to get an approximation of the PDEs in the form of algebraic equations that we can solve. In most practical cases, these equations cannot be solved analytically. The system of PDEs that describes these laws is nonlinear. The most accurate way to describe these laws is by partial differential equations (PDEs).
This eliminates the hassle associated with creating coarse meshes for complex geometries with small details that are difficult to mesh unless a fine mesh is used.Ī Fluid Flow Problem, Mathematical Model, and Numerical Modelįluid flow can be accurately described by the laws for conservation of momentum, mass, and energy. Available as of version 5.3a of the COMSOL Multiphysics® software, the AMG method only requires one mesh, in contrast to the geometric multigrid (GMG) solver, which requires at least one extra coarser mesh. The algebraic multigrid (AMG) solver provides robust solutions for large CFD simulations.
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