Provable convergence of blow-up time of numerical approximations for a class of convection-diffusion equations

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In this paper, we investigate the numerical algorithms to capture the blow-up time for a class of convection-diffusion equations with blow-up solutions. The numerical methods for such equations may not be straightforward to construct due to the lack of stability. Moreover, the blow-up time is more difficult to capture since we cannot distinguish whether the blow-up is physical or is due to the instability of the numerical methods. In this paper, we consider a class of convection-diffusion equations with positive blow-up solutions and the blow-up is due to the formation of δ-singularities. We use the positivity-preserving technique to enforce the L1-stability and the L2-norm of the numerical approximations to detect the blow-up phenomenon. We propose two ways to define the numerical blow-up time and prove their convergence to the exact one. As an application, we extend this method to calculate when the shock appears for scalar hyperbolic conservation laws. Three model problems will be discussed and tested to confirm the convergence numerically. Finally, the method can also be used to test whether an equation has a blow-up solution in finite time.

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Journal of Computational Physics