Adaptive semi-discrete formulation of BSQI–WENO scheme for the modified Burgers’ equation

In this article, we have proposed a septic B-spline quasi-interpolation (SeBSQI) based numerical scheme for the modified Burgers’ equation. The SeBSQI scheme maintains eighth order accuracy for the smooth solution, but fails to maintain a non-oscillatory profile when the solution has discontinuities or sharp variations. To ensure the non-oscillatory profile of the solution, we have proposed an adaptive SeBSQI (ASeBSQI) scheme for the modified Burgers’ equation. The ASeBSQI scheme maintains higher order accuracy in the smooth regions using SeBSQI approximation and in regions with discontinuities or sharp variations, 5th order weighted essentially non-oscillatory (WENO) reconstruction is used to preserve a non-oscillatory profile. To identify discontinuous or sharp variation regions, a weak local truncation error based smooth indicator is proposed for the modified Burgers’ equation. For the temporal derivative, we have considered the Runge–Kutta method of order four. We have shown numerically that the ASeBSQI scheme preserves the convergence rate of the SeBSQI and it converges to the exact solution with convergence rate eight. We have performed numerical experiments to validate the proposed scheme. The numerical experiments demonstrate an improvement in accuracy and efficiency of the proposed schemes over the WENO5 and septic B-spline collocation schemes. The ASeBSQI scheme is also tested for one-dimensional Euler equations.