6.1 Summary
In this dissertation, we have introduced a novel approximation system which we call the alternating evolution (AE) approximation system for one- and two-dimensional hyperbolic conservation laws and one-dimensional Hamilton-Jacobi equations. The numerical method is devised based on the AE approximation system by sampling over alternating computational grid points. The nature of solutions having singularities, which is generic to these equations in handled using the AE methodology. We formulate the first, second and third order accurate schemes and theoretical numerical stability is proved mainly for the first and second order schemes of hyperbolic conservation law and Hamilton-Jacobi equations. In the case of hyper-bolic conservation law we have also shown that the numerical solutions converge to the weak solution.
The designed methods have the advantage of being Riemann solver free, and the performs comparably to the finite volume/difference methods currently used. A series of numerical tests illustrates the capacity and accuracy of our method in describing the solutions.
6.2 Future work
Some future directions are outlined below:
• Local AE schemes for 2D conservation law: Many practical applications involve systems in two and three dimensions. More thorough study and analysis of the AE method-ology for 2D conservation law is underway which includes the formulation of local AE schemes. The local AE schemes for two-dimensional case will be an extension from the
one-dimensional case and would involve embedding local information into the parameter ǫ, and defining four local parameters in the four cells IijSE, IijN E, IijSW and IijN W that make up the computational cell Iij.
• Global and local AE schemes for 2D Hamilton-Jacobi equations: The AE scheme can also be formulated for two-dimensional Hamilton-Jacobi equations given by
φt+ H(φx, φy) = 0.
The AE system will be a natural extension from the one-dimensional AE system and propose to be of the form
ut+ Hvx, vy) = 1
ǫ(v − u), vt+ H(ux, uy) = 1
ǫ(u − v).
The numerical scheme is formulated using this approximate system.
• Applications to other systems A general convection-diffusion equation has the form
φt+ ∇ · f (φ) = ∇ · Q(φ, ∇φ),
f (φ) is a non-linear convection flux, and Q(φ, ∇φ) is a dissipation flux. These equations also arise in two phase flow in oil reservoirs, front propagation, traffic flow, financial modeling, and several other areas. We plan to formulate the AE schemes to these kind of equations.
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