This paper by Kurt J. Marfurt, published in Geophysics in 1984, quantitatively compares the accuracy of finite-difference and finite-element methods for solving the scalar and elastic wave equations. The study evaluates various time-domain and frequency-domain techniques, including explicit and implicit schemes, and assesses their performance in homogeneous and inhomogeneous media. Key findings include: - In homogeneous media, explicit finite-element and finite-difference schemes are comparable for solving the scalar wave equation and elastic wave equations with Poisson's ratio less than 0.3. - Finite-elements outperform finite-differences when modeling elastic media with Poisson's ratio between 0.3 and 0.45. - Implicit time integration schemes, such as the Newmark method, are inferior to explicit central-differences schemes, as they yield stable but inaccurate results for time steps exceeding the Courant condition. - Frequency-domain finite-element solutions using a weighted average of consistent and lumped masses provide the most accurate results and are cost-effective for CDP, well log, and interactive modeling. The paper also discusses the costs and limitations of different numerical methods, emphasizing that cost should not be the sole criterion for choosing an algorithm. It concludes that the unconditionally stable frequency-domain finite-element method with a weighted mass matrix is the most accurate and cost-effective for a fixed level of accuracy.This paper by Kurt J. Marfurt, published in Geophysics in 1984, quantitatively compares the accuracy of finite-difference and finite-element methods for solving the scalar and elastic wave equations. The study evaluates various time-domain and frequency-domain techniques, including explicit and implicit schemes, and assesses their performance in homogeneous and inhomogeneous media. Key findings include: - In homogeneous media, explicit finite-element and finite-difference schemes are comparable for solving the scalar wave equation and elastic wave equations with Poisson's ratio less than 0.3. - Finite-elements outperform finite-differences when modeling elastic media with Poisson's ratio between 0.3 and 0.45. - Implicit time integration schemes, such as the Newmark method, are inferior to explicit central-differences schemes, as they yield stable but inaccurate results for time steps exceeding the Courant condition. - Frequency-domain finite-element solutions using a weighted average of consistent and lumped masses provide the most accurate results and are cost-effective for CDP, well log, and interactive modeling. The paper also discusses the costs and limitations of different numerical methods, emphasizing that cost should not be the sole criterion for choosing an algorithm. It concludes that the unconditionally stable frequency-domain finite-element method with a weighted mass matrix is the most accurate and cost-effective for a fixed level of accuracy.