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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
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.
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.
The readings, questions and flashcards were written for this dataset. The linked Euler's method videos are by Khan Academy.
More to read
Building a heat update
A heat calculation starts with values at chosen positions and times. For u_t = k u_xx, take k > 0 and positive grid steps dx and dt. At one interior point, call the old left, centre and right values L, C and R. Replace the time derivative by (C_new - C)/dt and the second space derivative by (L - 2C + R)/dx^2. Solving for C_new gives C_new = C + r(L - 2C + R), where r = k dt/dx^2.
The update uses known old values, so it is explicit. Rewriting it gives weights r, 1 - 2r and r. They add to 1, which preserves a constant profile. A weighted-average interpretation also requires nonnegative weights. When r exceeds 0.5, the centre weight is negative and that interpretation no longer applies.
Checking the growth of grid noise
For periodic one-dimensional heat stepping, examine a Fourier mode with phase change theta between neighbouring points. FTCS multiplies it by G = 1 - 4r sin^2(theta/2). The squared sine ranges from 0 to 1. Requiring every amplification factor to lie between -1 and 1 gives 0 <= r <= 0.5. The alternating mode reaches G = -1 at r = 0.5, so it changes sign without decaying.
The restriction becomes dt <= dx^2/(2k). Halving dx quarters the largest allowed time step. Backward Euler instead uses new-time spatial values and requires a linear-system solve. Its factor is 1/(1 + 4r sin^2(theta/2)), with magnitude at most 1 for every nonnegative r. This removes the FTCS stability cap, but it does not make large steps accurate.
Three different checks
Consistency checks the residual obtained by inserting a smooth exact solution into a numerical scheme. Taylor expansion gives FTCS a leading time residual proportional to dt and a leading space residual proportional to dx^2. These are first-order time and second-order space consistency.
Stability asks for a bound on perturbation growth that remains uniform as the mesh is refined over a fixed time interval. Convergence asks whether the grid solutions approach the exact solution. For a consistent linear finite difference scheme for a well-posed linear initial-value problem, the Lax equivalence theorem relates stability and convergence in a specified norm. An exact solution can provide a useful numerical benchmark. Establishing consistency order by Taylor expansion does not require an exact solution formula.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Nothing; this is a starting point.
Optional connections
This opens up
Nothing builds on it yet.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.