Completely discrete schemes for 2D Sobolev equations with Burgers' type nonlinearity

Inthis paper, we discuss two first-order completely discrete schemes based on Backward Euler and its linearized variant methods for the 2D Sobolev equations with Burgers' type nonlinearity. First, we derive some a priori estimates for the semi-discrete scheme, then a priori bounds for the fully discrete scheme are obtained for the backward Euler approximation. Use of discrete Gronwall's Lemma and Stolz-Cesaro's classical result for sequences show that these estimates for the fully discrete scheme are valid uniformly in time. Moreover, an existence of a global attractor of a discrete dynamical system is derived. Further, optimal a priori error bounds are established, which may depend exponentially on time. It is shown that these error estimates are uniform in time under a uniqueness condition. Moreover, as the coefficient of dispersion mu in - mu Delta u(t) tends to zero, both the semi-discrete and completely discrete Sobolev equations converge to the corresponding Burgers' equation linearly with respect to mu. Finally, some numerical examples are established in support of our theoretical analysis.