Spline-in-compression and geometric meshes non-standard finite-difference compact discretization for solving Burgers type nonlinear parabolic PDEs

  • Navnit Jha South Asian University
  • Madhav Wagley Florida A\&M University, Tallahassee, FL


We describe a family of compact schemes for nonlinear parabolic partial differential equations using a spline-in-compression basis and geometric mesh network. The proposed scheme falls under the range of increased accuracy and is listed among the unconditional stable category. The numerical scheme is formulated with three non-uniformly spaced mesh points in the spatial direction and two uniformly space meshes on the time axis. Each time level yields a tridiagonal matrix system that makes computing more efficient in memory space. The interplay of parameters concerning the geometric meshes and spline-in-compression basis results in a more precise numerical solution value; this happens because one can assign the values of tuning parameters according to the position of layer concentration. The stability for the spline-in-compression high-order compact scheme is scrutinized with Fourier-based analysis. The scheme accomplishes an accuracy of third-order, and in a special case, it is fourth-order accurate. The numerical simulations with the Burger-Huxley equations, Burger-Fisher Equations, diffusion boundary-layer equations, Ilkovi\v{c}'s equation, and convection-diffusion equations are performed. The metric of estimates in terms of maximum absolute errors and computational convergence order supports the stability analysis.