aspg-33-4237-1792223308 20261017074828 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 2025 12 2 Zahraa Ahmed Sahib Islamic Azad University Tehran Branch, Iran Najmeh Malek Mohammadi Islamic Azad University Tehran Branch, Iran Normal false false false TR X-NONE AR-SA DefSemiHidden="false" DefQFormat="false" DefPriority="99" LatentStyleCount="371"> UnhideWhenUsed="true" QFormat="true" Name="heading 2"/> UnhideWhenUsed="true" QFormat="true" Name="heading 3"/> UnhideWhenUsed="true" QFormat="true" Name="heading 4"/> UnhideWhenUsed="true" QFormat="true" Name="heading 5"/> UnhideWhenUsed="true" QFormat="true" Name="heading 6"/> UnhideWhenUsed="true" QFormat="true" Name="heading 7"/> UnhideWhenUsed="true" QFormat="true" Name="heading 8"/> UnhideWhenUsed="true" QFormat="true" Name="heading 9"/> Name="index 1"/> Name="index 2"/> Name="index 3"/> Name="index 4"/> 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Accent 4"/> Name="Grid Table 7 Colorful Accent 4"/> Name="Grid Table 1 Light Accent 5"/> Name="Grid Table 6 Colorful Accent 5"/> Name="Grid Table 7 Colorful Accent 5"/> Name="Grid Table 1 Light Accent 6"/> Name="Grid Table 6 Colorful Accent 6"/> Name="Grid Table 7 Colorful Accent 6"/> Name="List Table 1 Light Accent 1"/> Name="List Table 6 Colorful Accent 1"/> Name="List Table 7 Colorful Accent 1"/> Name="List Table 1 Light Accent 2"/> Name="List Table 6 Colorful Accent 2"/> Name="List Table 7 Colorful Accent 2"/> Name="List Table 1 Light Accent 3"/> Name="List Table 6 Colorful Accent 3"/> Name="List Table 7 Colorful Accent 3"/> Name="List Table 1 Light Accent 4"/> Name="List Table 6 Colorful Accent 4"/> Name="List Table 7 Colorful Accent 4"/> Name="List Table 1 Light Accent 5"/> Name="List Table 6 Colorful Accent 5"/> Name="List Table 7 Colorful Accent 5"/> Name="List Table 1 Light Accent 6"/> Name="List Table 6 Colorful Accent 6"/> Name="List Table 7 Colorful Accent 6"/> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:8.0pt; mso-para-margin-left:0in; line-height:107%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri",sans-serif; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:Arial; mso-bidi-theme-font:minor-bidi; mso-ansi-language:TR;} This study focuses on the development of efficient numerical algorithms for solving nonlinear partial differential equations (PDEs). The research integrates theoretical analysis and practical numerical experiments to address the challenges posed by nonlinear PDEs, which often lack closed-form solutions and exhibit sensitivity to initial and boundary conditions. Benchmark models such as Burgers’ Equation, the Korteweg–de Vries (KdV) Equation, and the Navier–Stokes Equations are highlighted due to their significance in physical and engineering applications. Traditional numerical methods—Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM)—are reviewed with respect to accuracy, stability, and computational efficiency. Numerical stability concepts, including Von Neumann analysis and the CFL condition, are discussed alongside sources of error and strategies for error reduction. New algorithms were proposed by enhancing traditional schemes, incorporating adaptive mesh refinement, and integrating stability techniques. Numerical experiments on benchmark problems demonstrated improved accuracy, enhanced stability in handling nonlinear terms, and acceptable computational efficiency. The findings emphasize the importance of selecting suitable numerical methods, conducting stability analysis, and applying adaptive techniques. The study recommends higher-order schemes, conservative formulations for fluid dynamics, and double precision when necessary, ensuring reliable and reproducible computational results. 2025 40 50 4237 https://creativecommons.org/licenses/by/4.0/ 10.54216/GJMSA.120203 https://americaspg.com/journal/33/article/4237 https://americaspg.com/storage/51767022911.pdf