A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.

中文翻译：

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具有不连续初始条件的奇摄动对流扩散问题的数值逼近

研究了具有不连续初始条件的对流扩散型奇摄动抛物线问题。确定了一个解析函数，该函数与初始条件下的不连续性相匹配，并且还满足与该问题相关的齐次抛物线微分方程。该解析函数与抛物线问题解之间的差异通过使用迎风有限差分算子与适当的层自适应网格相结合来数值近似。数值方法显示为参数统一的。数值结果表明了本文建立的理论误差范围。