A modally optimized dynamic explicit nonlinear finite difference scheme: MODEN FDS
We present an Explicit, Non-linear, Dynamic Finite Difference Scheme (FDS) that is both Frequency and Amplitude sensitive. We refer to this scheme as MODEN FDS, i.e., Modally Optimized Dynamic Explicit Non-linear Finite Difference Scheme. Our MODEN FDS is designed for solving nonlinear problems and we demonstrate it on the Burgers equation. In order to control non-linear frequency and amplitude errors, we allow the Courant number (σ) to vary according to the maximum velocity at each time step. In conjunction with this we introduce a parameter vector [Special characters omitted.] , whose element is a fitting parameter for the finite difference approximation itself. Dependent on the selected σ, we choose an appropriate ν to reduce model errors at dominant frequencies. We compute frequency accurate and amplitude sensitive approximations over a range of σ. The results of these fitting coefficients are stored in Look-Up Tables, which are then used during the nonlinear simulation efficiently. For integration we use an optimized adaptive time increment. A Conservative Dynamic digital filter is designed and used after the integration step to control amplitude growth. The non-linear method appears to be stable, based on numerical experiments. Numerical results are compared with either theoretical solutions or results of other existing schemes. The overall performance of the Dynamic scheme maintains high fidelity, i.e., it contains essentially no error in both Frequency & Amplitude. The method is not a black box since it requires significant a priori work and a good understanding of the problem being solved. However, once developed, it is computationally efficient. Hence, the scheme is very efficacious.