--- title: "Numerical Schemes for Evolution Equations" description: "Replace derivatives by finite differences to compute successive time levels. For well-posed linear initial-value problems, consistency and stability give convergence. Explicit heat stepping restricts " canonical: https://lightmysky.com/learn/mathematics/numerical-schemes-for-evolution-equations-mt_RuDbOapexB source: https://lightmysky.com/learn/mathematics/numerical-schemes-for-evolution-equations-mt_RuDbOapexB.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Numerical Schemes for Evolution Equations Replace derivatives by finite differences to compute successive time levels. For well-posed linear initial-value problems, consistency and stability give convergence. Explicit heat stepping restricts the time step; backward Euler avoids that stability restriction. Subject: Mathematics · Area: Differential Equations · Ages 23 to 24 Page: https://lightmysky.com/learn/mathematics/numerical-schemes-for-evolution-equations-mt_RuDbOapexB ## Ready when they can - Identifies how derivative quotients produce FTCS, applies finite difference formulas, and states consistency orders - Tests Fourier amplification factors and obtains time-step restrictions - Explains mesh-ratio scaling and the roles of consistency and stability in convergence ## Lesson: Turning change into steps Green's functions above solved a spatial boundary value problem. Here the one-dimensional heat equation u_t = k u_xx, k > 0, evolves an initial profile. On a uniform grid, dx > 0 is spatial spacing, dt > 0 is the time step, and u_j^n approximates u at point j and time level n. Discretising space gives an ODE system; forward Euler uses old-time rates. The Fourier tests use periodic boundaries: grid values repeat after one spatial period. Write L, C and R for three neighbouring old values. Approximate u_t by (C_new - C)/dt and u_xx by (L - 2C + R)/dx^2. Substitution gives C_new = C + r(L - 2C + R), where r = k dt/dx^2. This is forward-time, centred-space stepping, or FTCS. Its weights are r, 1 - 2r and r. They sum to 1, so a constant profile stays constant. *(drawing: The weights form an average only when all three are nonnegative.)* **Example.** Take k = 1, dt = 0.002 and dx = 0.1. Then r = 0.002/0.01 = 0.2. For L = 2, C = 3 and R = 7, the update is 3 + 0.2(2 - 6 + 7) = 3.6. Equivalently, 0.2(2) + 0.6(3) + 0.2(7) = 3.6. At r > 0.5 the centre weight is negative, so this is no longer a weighted average. For a smooth solution, Taylor expansion gives the forward time quotient as u_t + (dt/2)u_tt + O(dt^2). The centred second space quotient is u_xx + (dx^2/12)u_xxxx + O(dx^4). Thus the FTCS residual has order O(dt + dx^2): first order in time and second order in space. These consistency orders do not by themselves establish convergence. A centred first derivative uses (f(x+h) - f(x-h))/(2h), with h > 0. The denominator is the distance between the sample points. Taylor expansion gives f'(x) + O(h^2). For f(x) = x^2 the quotient is exactly 2x: expanding the numerator gives 4xh, and dividing by 2h gives 2x. *(drawing: For smooth solutions, the leading residual terms scale like dt and dx squared.)* A Fourier grid mode is a sine or cosine pattern sampled on the periodic grid, with phase change theta per grid step. One FTCS step multiplies its amplitude by G = 1 - 4r sin^2(theta/2). Since sin^2 ranges from 0 to 1, |G| <= 1 for all phases requires -1 <= 1 - 4r <= 1, or 0 <= r <= 0.5. The alternating mode has G = 1 - 4r. At r = 0.5 it changes sign without shrinking: stability permits bounded oscillation. The FTCS bound gives dt_max = dx^2/(2k). Halving dx quarters this cap. If dt and k stay fixed, halving dx multiplies r by 4; crossing the bound depends on the old r. Doubling dx while quartering dt divides r by 16. To keep r fixed when dx doubles, multiply dt by 4. *(drawing: With k fixed, this change makes alternating grid noise grow by a factor of 3.8 per update.)* Consistency means the residual tends to zero on smooth exact solutions. Stability bounds perturbation growth on a fixed time interval uniformly as the mesh is refined. Convergence means grid solutions approach the exact solution. For a well-posed linear initial-value problem and a consistent linear finite difference scheme, the Lax equivalence theorem makes stability equivalent to convergence in the specified norm. Backward Euler uses new-time spatial values: C_new = C + r(L_new - 2C_new + R_new). It requires a linear-system solve each step. For periodic heat stepping, G = 1/(1 + 4r sin^2(theta/2)), so |G| <= 1 for every r >= 0. The constant mode has G = 1. This unconditional stability does not guarantee small error with large steps. Its consistency orders are first in time and second in space. **Recap.** Derive the update, check its consistency order and bound error growth. FTCS heat stepping needs r <= 0.5; backward Euler has no such time-step stability cap. *This lesson, question bank and flashcards were written for this dataset.* ## Practice 18 questions on this page, each with its working shown. ## Optional connections - [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR) - [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL)