Lemma 2.1 (Gradient of truncations, [8]).Let u be measurable onΩwith T k ( u ) ∈ W 0 1 , p ( Ω ) for every k > 0 . Then there exists a unique measurable function v : Ω → ℝ N such that

∇ T k ( u ) = v 1 { | u | < k } a.e., ∀ k > 0.

Moreover, if u ∈ W 0 1 , 1 ( Ω ) then v = ∇ u .

Lemma 2.2 ([9][10]). Let M : ℝ → ℝ be continuous and monotone with M ( 0 ) = 0 . Suppose u ∈ L 2 ( 0 , T ; H 0 1 ( Ω ) ) with M ( u ) ∈ L 1 ( Q T ) , ∂ t M ( u ) ∈ L 2 ( 0 , T ; H − 1 ( Ω ) ) + L 1 ( Q T ) , and u 0 ∈ L 1 ( Ω ) . Then for every h ∈ C c ∞ ( ℝ ) with h ( 0 ) = 0 , η ∈ H 0 1 ( Ω ) , and φ ∈ C c ∞ ( Q T ) with φ ( T ) = 0 ,

For ψ : ℝ → ℝ continuous and strictly increasing with ψ ( 0 ) = 0 , and M k ( r ) = M ( r ) + 1 k r (strictly increasing), define the operator

on the domain { u ∈ H 0 1 ( Ω ) ∩ L ∞ ( Ω ) : div   a ( u , ∇ u ) ∈ L ∞ ( Ω ) } , where a ( u , ∇ u ) = ∇ u + K ( u ) e z .

Proposition 2.3([6][8]).Under(H1)-(H2), A M k ψ is m -accretive and densely defined in L 1 ( Ω ) . Moreover, as M k → M uniformly on compacts, A M ψ ⊂ lim _ k → ∞ A M k ψ .

By nonlinear semigroup theory [11], Proposition 2.3 yields a unique mild solution v k = M k ( u k ) ∈ C ( [ 0 , T ] ; L 1 ( Ω ) ) of the Cauchy problem v ˙ k + A M k ψ ( v k ) ∋ f , v k ( 0 ) = v 0 k . As k → ∞ and v 0 k → v 0 in L 1 ( Ω ) , we have v k → v = M ( u ) in C ( [ 0 , T ] ; L 1 ( Ω ) ) .

Definition 3.1 (Renormalized solution).A measurable function u : Q T → ℝ is a renormalized solution of(1) if

(i) M ( u ) ∈ L 1 ( Q T ) ,

(ii) T k ( u ) ∈ L 2 ( 0 , T ; H 0 1 ( Ω ) ) for every k > 0 ,

(iii) for every h ∈ C c 1 ( ℝ ) and φ ∈ C c ∞ ( ( 0 , T ) × Ω ) ,

(iv) the following energy condition at infinity holds:

Remark 3.2.Condition (3) selects the physical solution among (possibly infinitely many) weak solutions: it expresses that no energy is dissipated at infinite amplitude levels. When supp   h ⊂ [ − k , k ] , the term ( ∇ u + K ( u ) e z ) ⋅ ∇ ( h ( u ) φ ) is identified with ( ∇ T k ( u ) + K ( T k ( u ) ) e z ) ⋅ ∇ ( h ( T k ( u ) ) φ ) , which is well-defined by (ii).

Lemma 3.3(Monotone comparison).Let u 0 , u ˜ 0 , f , f ˜ ∈ L ∞ ( Q T ) and let ψ , ψ ˜ : ℝ → ℝ be continuous and strictly increasing with ψ ( 0 ) = ψ ˜ ( 0 ) = 0 . Let u , u ˜ be weak solutions of the perturbed problem(1) with perturbations ψ and ψ ˜ respectively. If u 0 ≤ u ˜ 0 a.e.onΩ, f ≤ f ˜ a.e.on Q T ,and ψ ˜ ( r ) ≤ ψ ( r ) for all r ∈ ℝ , then u ≤ u ˜ a.e.on Q T .

Proof. Taking 1 L φ ( t ) ρ p ( t − s ) T L ( u ( t , x ) − u ˜ ( s , x ) ) as test function and using the identity

together with the same doubling argument as in Lemma 2.2 and Proposition 2.3, one obtains ∫ Q T ( ψ ˜ ( u ) − ψ ˜ ( u ˜ ) ) + d x d t ≤ 0 . Since ψ ˜ is strictly increasing, u ≤ u ˜ a.e. □

Theorem 3.4(Existence).Under(H1)-(H3),there exists at least one renormalized solution of(1).

Proof.Step 1: Approximation. Set f m , n = max ( min ( f , m ) , − n ) , u 0 , m , n = max ( min ( u 0 , m ) , − n ) , and ψ m , n ( r ) = 1 m r + − 1 n r − for m , n ∈ ℕ * . By Proposition 2.3 and nonlinear semigroup theory (applied with ψ = ψ m , n , which is strictly increasing), there exists a unique weak solution u m , n of the perturbed problem

Step 2: Monotonicity and pointwise limits. By Lemma 3.3,

where u n and u are measurable and almost-everywhere finite (see [6]).

Step 3: Energy estimate. Choosing test function T k ( u m , n ) φ δ in (4), where φ δ ∈ C 1 ( [ 0 , T ] ) is a cut-off near T , using Lemma 2.2 and the monotonicity of ψ m , n , one obtains

By the Poincaré inequality, ( T k ( u m , n ) ) is bounded in L 2 ( 0 , T ; H 0 1 ( Ω ) ) , so T k ( u m , n ) ⇀ T k ( u ) weakly in L 2 ( 0 , T ; H 0 1 ( Ω ) ) .

Step 4: Strong convergence of truncations. Taking h ( u n ) ( T k ( u n ) − T k ( u ) ) φ as test function in the equation satisfied by u n , one shows that I 1 + I 2 + I 3 → 0 (where I 1 involves the time derivative, I 2 the gradient cross-terms, and I 3 the convection), yielding ‖ T k ( u n ) ‖ L 2 ( H 0 1 ) → ‖ T k ( u ) ‖ L 2 ( H 0 1 ) . Combined with the weak convergence, this gives

Step 5: Passage to the limit. Using the strong convergence (6) and the Lebesgue dominated convergence theorem, one passes to the limit in each term of the equation for u n as n → ∞ to obtain (2).

Step 6: Energy condition at infinity. Taking ( T k + 1 ( u n ) − T k ( u n ) ) φ as test function and letting n → ∞ , then k → ∞ with φ = 1 , yields (3). □

Theorem 3.5(Uniqueness and comparison).Under(H1)-(H3),for i = 1 , 2 let u 0 i ∈ L 1 ( Ω ) , f i ∈ L 1 ( Q T ) , and u i be renormalized solutions of (1) with data ( u 0 i , f i ) . Then, for all t ∈ ( 0 , T ) ,

In particular, u 1 = u 2 a.e. when u 01 = u 02 and f 1 = f 2 .

Proof sketch. The proof uses the double-variable technique of [7], exploiting the structural condition

which holds for a ( u , ∇ u ) = ∇ u + K ( u ) e z since K is Lipschitz. A doubling of both the space and time variables, combined with the energy condition (3), yields (7). □

Let Ω = ] 0 , L [ with L = 1   m . We construct a mesh T h = { L 1 , ⋯ , L N + 1 } with nodes 0 = x 0 < x 1 < ⋯ < x N + 1 = L and h = max i ( x i + 1 − x i ) . The conforming finite-element space is

with nodal basis { φ j } j = 1 N . The semi-discrete problem reads: find u h ( t ) = ∑ m = 1 N u m ( t ) φ m ∈ V h such that

With time step τ = T / N t and u h k ≈ u h ( ⋅ , t k ) , the implicit Euler step gives:

Since (9) is nonlinear in u k , we linearise via a first-order Taylor expansion (Picard method):

where C j k , i − 1 = ∂ u { M , K } | u k , i − 1 . This yields, at each Picard step i , the linear system A k , i − 1 u k , i = B k , i − 1 with stiffness matrix

We work under the additional regularity hypotheses:

M is Lipschitz with M ′ ≥ m 0 > 0 on every compact (non-degenerate regime of Brooks-Corey [12]). u ∈ L ∞ ( 0 , T ; H 2 ( Ω ) ) ∩ H 1 ( 0 , T ; H 0 1 ( Ω ) ) ∩ H 2 ( 0 , T ; L 2 ( Ω ) ) . The Picard iteration converges in a uniformly bounded number N Picard of steps (guaranteed when τ is small relative to the Lipschitz constant of M ′ ).

Theorem 4.1(Convergence order). Under (H1)-(H6), there exists C > 0 independent of h and τ such that

Proof sketch. Decompose u ( ⋅ , t k ) − u h k = η k + θ k where η k = u ( ⋅ , t k ) − Π h u ( ⋅ , t k ) is the elliptic projection error and θ k = Π h u ( ⋅ , t k ) − u h k . By standard interpolation theory and H 2 -regularity of Ω = ] 0 , L [ ,

Testing the error equation for θ k with θ k and using the monotonicity of M (hypothesis (H4)) gives

Since θ 0 = 0 , we obtain (10). The estimate (11) follows analogously without the Aubin-Nitsche duality argument. □

Proposition 4.2(Unconditional stability). Under(H1)-(H2), for every τ > 0 and h > 0 ,

No CFL condition is required.

We validate the scheme on a one-dimensional infiltration problem with Brooks-Corey constitutive relations [12] (see Table 1 for the physical parameters and numerical settings):

Table 1. Brooks-Corey parameters and numerical settings.

Reduction to homogeneous Dirichlet data.The analytical model (1) is set with homogeneous Dirichlet boundary conditions u | ∂ Ω = 0 and the finite-element space V h enforces v h ( 0 ) = v h ( L ) = 0 . In the numerical experiment the physical boundary data are u ( 0 , t ) = 0 and u ( 1 , t ) = h c . To reconcile these with the analytical framework, we introduce the affine lifting

which satisfies u ˜ ( 0 , t ) = 0 and u ˜ ( 1 , t ) = 0 , so that u ˜ belongs to the homogeneous space V h for all t . The equation for u ˜ retains the same form as (1) with a modified right-hand side f ˜ = f + Δ ( h c x ) = f (since the lifting is affine), thereby preserving the structure of the analytical model. In the remainder of this section, u denotes the lifted variable u ˜ and all results of Sections 3-4 apply without modification.

Initial condition. The initial condition is chosen as

which satisfies u 0 ( 0 ) = 0 (surface at saturation) and u 0 ( 1 ) = h c (unsaturated base), consistently with the boundary data above. Its derivative u ′ 0 ( x ) = π 2 h c sin ( π x ) ≤ 0 reflects a physically realistic suction profile increasing with depth.

Figure 1 shows the evolution of the matric potential u ( x , t ) over ten time steps.

Figure 1. Evolution of the matric potential u ( x , t ) over ten time steps ( τ = 0.01   s ). The blue curve is the initial condition; as t → ∞ the profile tends uniformly to zero (fully saturated medium). Computed with Scilab on a mesh of N = 100 nodes.

The simulation exhibits three physically consistent features:

1) The initial profile (blue) displays a strong gradient near x = 1 , consistent with the prescribed base condition.

2) As t increases, u converges uniformly toward zero, reflecting the progressive saturation of the medium.

3) Convergence is rapid in the early steps and slows as the profile becomes quasi-uniform—a behaviour well-known from Richards equation theory.

Unconditional stability (Proposition 4.2) justifies the choice τ = h = 0.01 independently of any CFL constraint.