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The finite difference streamline diffusion methods for Sobolev equations with convection-dominated term. (English) Zbl 1030.65095

Consider the linear Sobolev equation \[ \begin{split} c(x,t){\partial u\over\partial t}+ d(x,t)\cdot\nabla u-\nabla\cdot \Biggl[a(x, t)\nabla u+ b(x,t) \nabla{\partial u\over\partial t}\Biggr]+ \sigma(x, t) u=\\ f(x, t),\qquad (x,t)\in \Omega\times (0,T]\end{split} \] with the boundary and initial conditions \[ u(x, t)= 0,\quad (x, t)\in\partial\Omega\times [0,T],\quad u(x,0)= u_0(x),\quad x\in\Omega,\quad t= 0,\quad\Omega\subset \mathbb{R}^2, \] with some smoothness and positivity conditions on the coefficients. The purpose of the present paper is to derive two “finite difference streamline diffusion schemes”, to study their stability and give error estimates in suitable norms.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:

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