This is the default equation for a Wave Form PDE interface. Here the coefficients for a wave form PDE are specified with the following equation coefficients:

Enter a value or expression for the damping (or mass) coefficient da. The default is 1. If there are multiple dependent variables, there is a matrix of da component inputs.

Enter values or expressions for the components of the conservative flux vector Γ(u). These components may depend on both the spatial and temporal coordinates, and the solution u, but not any derivatives of u. If there are multiple dependent variables, there is one Γ(u) vector for each variable.

Enter a value or expression for the source term f. If there are multiple dependent variables, there is a vector of f component inputs.

The shape functions used are discontinuous and therefore require auxiliary constraints on faces between adjacent mesh elements to yield a meaningful (that is, continuous) solution approximation. This is accomplished by specifying a so-called numerical flux on each face. The numerical flux implemented is the (global) Lax-Friedrichs flux, which is defined as the average of the fluxes on neighboring elements plus the jump of the solution times at parameter τ, which is necessary for stability. You can also specify a general numerical flux. From the Method list, choose Lax-Friedrichs (the default) to specify parameter τ, or choose General to specify a general numerical flux g*.

For Lax-Friedrichs, enter a value or global expression for the parameter τ. Only one expression can be entered for each equation and each domain. The parameter is by default one but should be set according to the dominant eigenvalue of the flux Jacobian matrix

where λ(da) are the eigenvalues of the mass matrix da. The reason for this extra factor is that the mass matrix inverse is applied to the Lax-Friedrichs flux internally. A so-called central flux is obtained for τ = 0. Selecting

Enter a value or expression for the estimate of maximum wave speed Ws. The default is 0.

The filter provides higher-order smoothing of nodal discontinuous Galerkin formulations and is intended to be used for absorbing layers, but you can also use it to stabilize linear wave problems with highly varying coefficients. The filter is constructed by transforming the solution (in each global time step) to an orthogonal polynomial representation, multiplying with a damping factor and then transforming back to the (Lagrange) nodal basis. Select the Activate check box to use this filter.

where V is a Vandermonde matrix induced by the node points, and Λ is a diagonal matrix with the exponential damping factors on the diagonal:

and Np is the basis function and im the polynomial order for coefficient m. α (default value: 36), ηc (default value: 0.6), and s (default value: 3) are the filter parameters that you specify in the corresponding text fields. The damping is derived from a spatial dissipation operator of order 2s. For s = 1, you obtain a damping that is related to the classical 2nd-order Laplacian. Higher order (larger s) gives less damping for the lower-order polynomial coefficients (a more pronounced low-pass filter), while keeping the damping property for the highest values of η, which is controlled by α. The default values 36 for a correspond to maximal damping for η = 1. It is important to realize that the effect of the filter is influenced by how much of the solution (energy) is represented by the higher-order polynomial coefficients. For a well-resolved solution this is a smaller part than for a poorly resolved solution. The effect is stronger for poorly resolved solutions than for well-resolved ones. This is one of the reasons why this filter is useful in an absorbing layer where the energy is transferred to the higher-order coefficients through a coordinate transformation. See Ref. 1 (Chapter 5) for more information.

α must be positive; α = 0 means no dissipation, and the maximum value is related to the machine precision, −log(ε), which is approximately 36. ηc should be between 0 and 1, where ηc = 0 means maximum filtering, and ηc = 1 means no filtering, even if filtering is active.

If desired, you can add a WENO (weighted essentially nonoscillatory) limiter (see The WENO Limiter) to control spurious oscillations around discontinuities and stabilize the computations. From the Limiter list, choose WENO to activate the WENO limiter. You can then also activate the TVB (total variation bounded) trouble cell indicator by choosing TVB from the Troubled cell indicator list and specify a TVB constant (default: 10). A larger value means that a smaller number of cells will use WENO.