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An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions
So-Hsiang Chou, Patrick Nyadjo Fonga
TL;DR
Nonlinear interface jumps depend on unknown solution traces, complicating standard immersed interface corrections. The paper combines Crank–Nicolson immersed interface discretization with an s-parameter scalar reduction and compares IIM–Newton with Newton–IIM. Experiments report second-order behavior, while trace recovery and the practical suitability of continuous Newton–IIM remain scope boundaries.
Problem
Nonlinear interface conditions make immersed interface corrections difficult because the jumps depend implicitly on unknown solution traces.
Method
The paper combines Crank–Nicolson immersed interface discretization with an s-parameter reduction and examines IIM–Newton and Newton–IIM formulations.
Results
The experiments show second-order convergence for the elliptic and parabolic formulations, with parabolic accuracy persisting for diffusion contrasts up to β+/β− = 250.
Takeaways & Limitations
The s-parameter reduction separates nonlinear interface coupling from linear bulk equations, enabling immersed interface discretization of auxiliary linear problems.
Takeaways & Limitations
The paper does not systematically study trace-recovery procedures, and continuous Newton–IIM is less natural within the standard immersed interface framework.
Abstract
from arXiv · showhide
We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition \[ [u]_α=λu^+u^-. \] The method combines a Crank--Nicolson immersed interface discretization with an \(s\)-parameter reduction of the nonlinear interface condition, thereby reducing the nonlinear coupling to a scalar quadratic equation. We also discuss two different viewpoints for combining Newton iteration with immersed interface discretization, namely the IIM--Newton and Newton--IIM formulations. Numerical experiments are presented to illustrate the behavior and accuracy of the method.
1. Introduction
The paper studies a one-dimensional nonlinear parabolic interface problem in which the jump depends on unknown solution traces. It incorporates this nonlinear condition into a classical immersed interface framework and compares two Newton-based formulations.
- Problem setting: Nonlinear jump conditions depend implicitly on unknown one-sided traces, making immersed interface correction terms harder to construct than for prescribed jumps.The interface law [u]_α = λu^+u^- couples the traces across the interface.
- Method: The method uses an s-parameter representation that reduces the nonlinear interface condition to a scalar quadratic equation.The decomposition produces two auxiliary linear problems, with u0 satisfying homogeneous jumps and u1 a unit solution jump.
- Scope and contribution: The paper investigates how nonlinear jump conditions can be incorporated into a classical immersed interface framework rather than introducing a new general nonlinear stencil.The study examines two viewpoints for combining Newton iteration with immersed interface discretization.
- Newton formulations: IIM–Newton discretizes first and then applies Newton iteration, localizing nonlinearity near irregular interface grid points while preserving sparse Jacobian corrections.This formulation applies Newton’s method to the resulting nonlinear algebraic system.
- Newton formulations: Newton–IIM linearizes the continuous nonlinear problem first, but the paper identifies practical difficulties in discretizing the resulting correction equation within the standard framework.The paper characterizes this formulation as less natural than IIM–Newton.
- Verification scope: Verification experiments use manufactured exact one-sided traces, while systematic recovery of traces from unfitted numerical data remains outside the paper’s scope.This setup isolates discretization and scalar-reduction effects from trace-reconstruction errors.
2. Immersed Interface Discretization for a Linear Problem
The linear immersed interface discretization uses standard centered differences away from the interface and Taylor-derived corrections at irregular grid points. Although local defects are lower order there, the discrete Green-function scaling supports second-order global accuracy.
- Discretization framework: The linear parabolic interface problem provides the building block for the nonlinear problem, with correction terms derived for prescribed interface jumps.The derivation records the correction terms needed at irregular grid points.
- Spatial discretization: A uniform mesh distinguishes two irregular grid points adjacent to the interface; standard centered approximations are used elsewhere.The stencil depends on whether a grid point is regular or irregular.
- Time discretization: The θ-method provides time discretization, with θ = 1/2 corresponding to Crank–Nicolson.For Crank–Nicolson, correction terms at consecutive time levels enter through their trapezoidal average.
- Correction terms: Taylor expansions about the interface determine correction terms at both irregular grid points by matching the interface conditions and derivative information.The construction proceeds separately on the two sides of the interface and simplifies the resulting corrections.
- Consistency and accuracy: Regular grid points have O(h^2) spatial truncation error, whereas matching through quadratic terms leaves O(h^3) unmatched expansion terms at irregular points before operator scaling.The second-difference coefficients are O(h^-2), producing lower local order at irregular points.
- Consistency and accuracy: O(h^2) global accuracy remains possible because O(h) defects occur at only two points and the discrete inverse contributes O(h) there; numerical results confirm this behavior.Regular-point truncation errors also contribute O(h^2) globally.
3. IIM–Newton Formulation: A Stationary Model Problem
The IIM–Newton formulation embeds the nonlinear jump law directly into localized immersed-interface corrections while retaining the linear bulk discretization. Newton iteration then modifies the sparse system only near the interface, subject to trace-reconstruction and denominator constraints.
- Nonlinear correction construction: The stationary model retains the linear immersed-interface construction by replacing the prescribed jump with a nonlinear function of the discrete solution.The nonlinear interface condition is treated as a solution-dependent jump in the irregular-point correction terms.
- Localized nonlinear structure: The finite-difference operator Ah remains unchanged, while all nonlinear dependence is collected in the correction vector Φ(U) with fixed local support.The correction affects only the irregular grid points, independently of the total number of grid points.
- Nonlinear correction construction: The nonlinear correction terms depend on the scalar quantity µ(U), which is computed from an approximation of the one-sided interface trace.The implementation uses quadratic same-side extrapolation from three neighboring minus-side grid values.
- Consistency and limitations: If the jump is prescribed independently of U, then Φ′(U)=0 and the formulation reduces to the linear immersed-interface scheme.This provides a direct consistency link between the nonlinear and prescribed-jump formulations.
- Localized nonlinear structure: Newton’s Jacobian differs from the linear immersed-interface matrix only through the localized derivative Φ′(U), preserving the sparse bulk structure.The derivative depends on only a few nearby grid values, so the nonlinear modification remains near the interface.
- Consistency and limitations: The Newton iteration requires a denominator to remain nonzero; otherwise, damping or restarting from another initial guess is recommended.A systematic analysis of alternative trace-recovery procedures is outside the paper’s scope.
4. Numerical Experiments
The experiments examine Newton iteration for a nonlinear elliptic interface problem and verify the Crank–Nicolson immersed interface scheme with s-parameter reduction for a nonlinear parabolic problem. The reported tests show second-order convergence, including short times and diffusion contrasts up to β+/β−=250.
- Nonlinear elliptic interface problem: The elliptic example has two exact solution branches, continuous uc and discontinuous ud, and Newton convergence depends on the initial iterate.The continuous branch appears to have a larger basin of attraction, while larger initial values may lead to the discontinuous branch.
- Nonlinear elliptic interface problem: The immersed interface approximations reproduce both the discontinuous and continuous elliptic solutions.
- Nonlinear elliptic interface problem: Second-order convergence is observed in the maximum norm for the discontinuous elliptic solution.The same second-order behavior persists when β+/β−=250.
- Nonlinear parabolic interface problem: The parabolic scheme decomposes the solution as u(x,t)=u0(x,t)+s(t)u1(x,t), with s(t) determined by the nonlinear jump condition.The auxiliary traces used in the manufactured experiment are evaluated from exact auxiliary solutions.
- Nonlinear parabolic interface problem: At T=7, observed orders approach two very closely for the reconstructed nonlinear solution.Because s is recovered to essentially machine precision, the reported error primarily reflects the Crank–Nicolson immersed interface discretization.
- Nonlinear parabolic interface problem: At T=0.1, convergence rates approach two for both coefficient choices, including β+/β−=250, and close agreement is observed on both sides of the interface.The shorter-time test retains a significant transient component.
5. Remarks on the Newton–IIM Formulation
The Newton–IIM approach linearizes the nonlinear interface problem before immersed interface discretization, producing a correction problem whose coefficients and interface data depend on the current iterate. Residual evaluation near the interface requires derivative or extrapolation procedures, making this formulation less natural than IIM–Newton.
- Newton–IIM first linearizes the continuous nonlinear interface problem, then applies the immersed interface method to the resulting correction equation.The correction problem is linear parabolic, but its coefficients and interface data depend on the current iterate.
- The Newton correction residual involves derivatives of the current iterate u.
- Because the iterate is available only through discrete grid values, consistent residual evaluation near the interface requires additional differentiation or extrapolation.
- Continuous Newton–IIM is less natural within the standard immersed interface framework than IIM–Newton, which discretizes first and applies Newton to the nonlinear algebraic system.