The Time-Explicit Solver Algorithms
The Time-Explicit Solver Runge–Kutta and Adams–Bashforth methods are discussed in this section. For the nodal discontinuous Galerkin method, it is natural and most efficient to use an explicit time-stepping method. Other situations when it can be advantageous is when using only particle tracing or wave problems together with so-called mass lumping.
TimeExplicit in the COMSOL Multiphysics Programming Reference Manual.
Runge–Kutta methods
Explicit classical Runge–Kutta methods of order 1–4 are supported. Runge–Kutta 4 is the default choice for the discontinuous Galerkin method.
Adams-Bashforth methods
The third-order Adams-Bashforth multistep method (AB3) for ut = R(u) is
where un is the solution at time tn, and k is the time step.
The time restriction for the discontinuous Galerkin method for wave problems is directly proportional to the smallest mesh element size.
About the Wave Form PDE Interface
About Auxiliary Equation-Based Nodes is tailored toward explicit time stepping. The method is quadrature free as well as matrix free. Only element local matrices are formed. A suitable stable time step can be determined automatically by specifying the variable wahw.wtc, which should be an estimate of the maximum wave speed for the equations in the interface. The Time Explicit algorithm then translates this speed to a local so-called cell time scale. For a global time marching method like Runge–Kutta or Adams–Bashforth 3, the time step is directly related to the smallest cell time scale. When there is a large difference in cell time scales, a global time marching method is not very efficient. For this reason, there is also a local time marching method, Adams–Bashforth 3 (local), which divides the cells into groups based on the cell time scale. The groups are then time marched with different time-step sizes, making this a more efficient method.
The Runge–Kutta and Adams–Bashforth methods can handle couplings of the Wave Form PDE interface (or any interface that uses nodal discontinuous Lagrange elements) with any other interface. There are two restrictions:
•
The time-dependent PDE problem must be first order in time with a linear dependency with respect to (see About the Time-Dependent Solver for the notation).
•
If the finite element discretization leads to a DAE, then its index must be 1 (see the Glossary). The algebraic equations are solved using a Fully Coupled solver. The frequency with which these algebraic equations are solved can be controlled with the Algebraic equations setting, which makes it possible to reduce the computational time. For example, this setting is useful to reduce the cost of solving a wave problem (discretized using the nodal discontinuous Galerkin method) coupled to an elliptic PDE (see the Glossary).