The Wave Form PDE Interface

The Wave Form PDE interface (), found under the Mathematics>PDE Interfaces branch () when adding an interface, solves wave equations formulated with first order derivatives in time and space using the discontinuous Galerkin method and is highly optimized with respect to speed and memory consumption.

This section covers the formulation and settings pertaining to those equations.

When you add this interface, these default nodes are also added to the Model Builder: Wave Form PDE, Zero Flux, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Wave Form PDE to select features from the context menu.

Use the wave form for first-order hyperbolic PDEs. Assuming a scalar equation for the dependent variable u, these problems take the form

together with suitable initial data. The first equation is the PDE, the second the boundary conditions.

The terms da, Γ, f, and g are coefficients. They can be functions of both the spatial coordinates or time, and the solution u, but not the derivatives of u. The coefficients f and g are scalar, whereas Γ is the flux vector. The coefficient da is assumed to be nonzero throughout the domain Ω, and for all times.

The Wave Form PDE interface also supports systems of equations. The interpretation of the coefficients are the same as for the scalar case; f and g are vectors with one component for each equation. Γ contains one flux vector for each equation, and da is a square (invertible) matrix.

Settings

The Label is the default physics interface name.

The Name is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers, and underscores (_) are permitted in the Name field. The first character must be a letter.

The default Name (for the first Wave Form PDE interface in the model) is wahw.

Units

Select a Dependent variable quantity. Dimensionless (1) is the default and there are many other options to choose from. Select a Source term quantity. None is the default and there are many other options to choose from. Enter a Unit.

Discretization

To display this section, click the Show button () and select Discretization.

Due to efficiency reasons, there is only one Shape function type (finite element type) defined for this interface — the Nodal discontinuous Lagrange functions. The associated element order (the order of the shape function for the element) can be chosen from the Element order list. The highest available order is 10 in 1D and 2D models and 7 in 3D models, and the default order is two.

The Nodal Discontinuous Lagrange shape functions can only be defined on triangular meshes in 2D or tetrahedral meshes in 3D.

Dependent Variables

Define the number of the dependent variables and the variable names. The default name for a scalar PDE variable is u.

The dependent variables and their names must be unique with respect to all other dependent variables in the model.

Add or remove dependent variables in the model and also change their names.

Enter the Number of dependent variables. Use the Add dependent variable () and Remove dependent variable () buttons as needed.

• Domain and Boundary Physics for the Wave Form PDE Interface • Working with Geometric Entities • Theory for the Wave Form PDE • The Time Explicit Solver Algorithms • Using Units