本研究探討微分方程之最大值原理應用於有限差分法求解非線性偏微分方程式的較大、較小近似解上下限及數值解的誤差範圍。研究方法採用偏微分方程最大值原理為基礎建構方程式的單調殘差關係式，並以有限差分法離散統御方程式，及搭配殘差修正法，求得拘束條件下的不等式最佳解，並將殘差修正的程序併入有限差分法非線性疊代過程中更可快速地求解相對於正確解的較大與較小近似解，至於在非線性問題的求解上無需增加額外的疊代次數即可求得近似解的最大誤差範圍。而根據求得的較大及較小暫態近似解不但能夠正確分析數值解與解析解的最大可能誤差範圍，數值驗證也顯示求得之近似解上下限的平均值仍具有良好的精度。
因此，殘差修正法具備誤差分析的特性，並能有效解決傳統數值方法上盲目的增加格點之缺點，達到節省計算時間、降低記憶體空間與避免不必要的反覆測試。整體來說，殘差修正法為便利且效率高並具有良好精度的數值方法。

This study investigates the method of the maximum principle applying on differential equations to solve the upper and lower approximate solutions of non-linear partial differential equations and their error range. Under the fundamental of the maximum principle, we establish the monotonic residual relations of the partial differential governing equations. We use the finite difference method to discretize the equations, and combine the Residual Correction Method to obtain the optimal solutions under the constrain of inequalities. Incorporating the Residual Correction Method into the nonlinear iteration procedure of the finite difference, the comparisons between the upper/lower approximate solutions and the exact solution can be achieved effectively. For the nonlinear problem, it is unnecessary to add additional iteration times and obtain the range of the maximum possible error with the analytic solutions. With respect to the upper and lower transient approximate solution, we can precisely analyze the range of the maximum possible error with the analytic solutions. The results reveal that the mean values of the upper and lower approximate solutions show the satisfactory accuracy.

Therefore, the Residual Correction Method with error-analysis characteristic can correct the defects resulting from increasing the numbers of grids or approximate functions when using traditional numerical methods. The methodology can save the computing time, reduce the storage of memory and avoid unnecessary repeated testing. Generally, the Residual Correction Method is an excellent numerical method. It can solve the varieties of complicated partial differential equations with preciseness and effectiveness.

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