WE EXTEND THE FINITE VOLUME NUMERICAL SCHEME PROPOSED BY HILLIGES AND WEIDLICH [11] TO SECOND ORDER TRAFFIC FLOW MODELS CONSISTING IN SYSTEMS OF NON STRICTLY HYPERBOLIC CONSERVATION LAWS OF TEMPLE CLASS. THE SCHEME IS SHOWN TO SATISFY SOME MAXIMUM PRINCIPLE PROPERTIES ON THE DENSITY. WE PROVIDE NUMERICAL TESTS ILLUSTRATING THE BEHAVIOUR AT VACUUM AND THE GAIN IN COMPUTATIONAL TIME WHEN DEALING WITH OPTIMIZATION ALGORITHMS.

THE SIMULATION MODEL PROPOSED IN [M. HILLIGES AND W. WEIDLICH. A PHENOMENOLOGICAL MODEL FOR DYNAMIC TRAFFIC FLOW IN NETWORKS. TRANSPORTATION RESEARCH PART B: METHODOLOGICAL, 29 (6): 407?431, 1995] CAN BE UNDERSTOOD AS A SIMPLE METHOD FOR APPROXIMATING SOLUTIONS OF SCALAR CONSERVATION LAWS WHOSE FLUX IS OF DENSITY TIMES VELOCITY TYPE, WHERE THE DENSITY AND VELOCITY FACTORS ARE EVALUATED ON NEIGHBORING CELLS. THE RESULTING SCHEME IS MONOTONE AND CONVERGES TO THE UNIQUE ENTROPY SOLUTION OF THE UNDERLYING PROBLEM. THE SAME IDEA IS APPLIED TO DEVISE A NUMERICAL SCHEME FOR A CLASS OF ONE-DIMENSIONAL SCALAR CONSERVATION LAWS WITH NONLOCAL FLUX AND INITIAL AND BOUNDARY CONDITIONS. UNIQUENESS OF ENTROPY SOLUTIONS TO THE NONLOCAL MODEL FOLLOWS FROM THE LIPSCHITZ CONTINUOUS DEPENDENCE OF A SOLUTION ON INITIAL AND BOUNDARY DATA. BY VARIOUS UNIFORM ESTIMATES, NAMELY A MAXIMUM PRINCIPLE AND BOUNDED VARIATION ESTIMATES, ALONG WITH A DISCRETE ENTROPY INEQUALITY, THE SEQUENCE OF APPROXIMATE SOLUTIONS IS SHOWN TO CONVERGE TO AN ENTROPY WEAK SOLUTION OF THE NONLOCAL PROBLEM. THE IMPROVED ACCURACY OF THE PROPOSED SCHEME IN COMPARISON TO SCHEMES BASED ON THE LAX-FRIEDRICHS FLUX IS ILLUSTRATED BY NUMERICAL EXAMPLES. A SECOND-ORDER SCHEME BASED ON MUSCL METHODS IS PRESENTED.

THE WELL-KNOWN LIGHTHILL-WHITHAM-RICHARDS (LWR) KINEMATIC MODEL OF TRAFFIC FLOW MODELS THE EVOLUTION OF THE LOCAL DENSITY OF CARS BY A NONLINEAR SCALAR CONSERVATION LAW. THE TRANSITION BETWEEN FREE AND CONGESTED FLOW REGIMES CAN BE DESCRIBED BY A FLUX OR VELOCITY FUNCTION THAT HAS A DISCONTINUITY AT A DETERMINED DENSITY. A NUMERICAL SCHEME TO HANDLE THE RESULTING LWR MODEL WITH DISCONTINUOUS VELOCITY WAS PROPOSED IN [J.D. TOWERS, A SPLITTING ALGORITHM FOR LWR TRAFFIC MODELS WITH FLUX DISCONTINUITIES IN THE UNKNOWN, J. COMPUT. PHYS., 421 (2020), ARTICLE 109722]. A SIMILAR SCHEME IS CONSTRUCTED BY DECOMPOSING THE DISCONTINUOUS VELOCITY FUNCTION INTO A LIPSCHITZ CONTINUOUS FUNCTION PLUS A HEAVISIDE FUNCTION AND DESIGNING A CORRESPONDING SPLITTING SCHEME. THE PART OF THE SCHEME RELATED TO THE DISCONTINUOUS FLUX IS HANDLED BY A SEMI-IMPLICIT STEP THAT DOES, HOWEVER, NOT INVOLVE THE SOLUTION OF SYSTEMS OF LINEAR OR NONLINEAR EQUATIONS. IT IS PROVED THAT THE WHOLE SCHEME CONVERGES TO A WEAK SOLUTION IN THE SCALAR CASE. THE SCHEME CAN IN A STRAIGHTFORWARD MANNER BE EXTENDED TO THE MULTICLASS LWR (MCLWR) MODEL, WHICH IS DEFINED BY A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR DRIVER CLASSES THAT ARE DISTINGUISHED BY THEIR PREFERENTIAL VELOCITIES. IT IS SHOWN THAT THE MULTICLASS SCHEME SATISFIES AN INVARIANT REGION PRINCIPLE, THAT IS, ALL DENSITIES ARE NONNEGATIVE AND THEIR SUM DOES NOT EXCEED A MAXIMUM VALUE. IN THE SCALAR AND MULTICLASS CASES NO FLUX REGULARIZATION OR RIEMANN SOLVER IS INVOLVED, AND THE CFL CONDITION IS NOT MORE RESTRICTIVE THAN FOR AN EXPLICIT SCHEME FOR THE CONTINUOUS PART OF THE FLUX. NUMERICAL TESTS FOR THE SCALAR AND MULTICLASS CASES ARE PRESENTED.

THIS STUDY REVISES THE NON-LOCAL MACROSCOPIC PEDESTRIAN FLOW MODEL PROPOSED IN [R. M. COLOMBO, M. GARAVELLO, AND M. LÉCUREUX-MERCIER. A CLASS OF NONLOCAL MODELS FOR PEDESTRIAN TRAFFIC. MATH. MODELS METHODS APPL. SCI., 22(4):1150023, 2012] TO ACCOUNT FOR ANISOTROPIC INTERACTIONS AND THE PRESENCE OF WALLS OR OTHER OBSTACLES IN THE WALKING DOMAIN. WE PROVE THE WELL-POSEDNESS OF THIS EXTENDED MODEL AND WE APPLY HIGH-RESOLUTION NUMERICAL SCHEMES TO ILLUSTRATE THE MODEL CHARACTERISTICS. IN PARTICULAR, NUMERICAL SIMULATIONS HIGHLIGHT THE ROLE OF DIFFERENT MODEL PARAMETERS IN THE OBSERVED PATTERN FORMATION.

IN THIS PAPER, WE PROPOSE AND ANALYZE A REACTION-DIFFUSION MODEL FOR PREDATOR-PREY INTERACTION, FEATURING BOTH PREY AND PREDATOR TAXIS MEDIATED BY NONLOCAL SENSING. BOTH PREDATOR AND PREY DENSITIES ARE GOVERNED BY PARABOLIC EQUATIONS. THE PREY AND PREDATOR DETECT EACH OTHER INDIRECTLY BY MEANS OF ODOR OR VISIBILITY FIELDS, MODELED BY ELLIPTIC EQUATIONS. WE PROVIDE UNIFORM ESTIMATES IN LEBESGUE SPACES WHICH LEAD TO BOUNDEDNESS AND THE GLOBAL WELL-POSEDNESS FOR THE SYSTEM. NUMERICAL EXPERIMENTS ARE PRESENTED AND DISCUSSED, ALLOWING US TO SHOWCASE THE DYNAMICAL PROPERTIES OF THE SOLUTIONS.

ONE-DIMENSIONAL MODELS OF GRAVITY-DRIVEN SEDIMENTATION OF POLYDISPERSE SUSPENSIONS WITH PARTICLES THAT BELONG TO N SIZE CLASSES GIVE RISE TO SYSTEMS OF N STRONGLY COUPLED, NONLINEAR FIRST-ORDER CONSERVATION LAWS FOR THE LOCAL SOLIDS VOLUME FRACTIONS. AS THE EIGENVALUES AND EIGENVECTORS OF THE FLUX JACOBIAN HAVE NO CLOSED ALGEBRAIC FORM, CHARACTERISTIC-WISE NUMERICAL SCHEMES FOR THESE MODELS BECOME INVOLVED. ALTERNATIVE SIMPLE SCHEMES FOR THIS MODEL DIRECTLY UTILIZE THE VELOCITY FUNCTIONS AND ARE BASED ON SPLITTING THE SYSTEM OF CONSERVATION LAWS INTO TWO DIFFERENT FIRST-ORDER QUASI-LINEAR SYSTEMS, WHICH ARE SOLVED SUCCESSIVELY FOR EACH TIME ITERATION, NAMELY, THE LAGRANGIAN AND REMAP STEPS (SO-CALLED LAGRANGIAN-REMAP [LR] SCHEMES). THIS APPROACH WAS ADVANCED IN (BÜRGER, CHALONS, AND VILLADA, SIAM J SCI COMPUT 35 (2013), B1341?B1368) FOR A MULTICLASS LIGHTHILL?WHITHAM-RICHARDS TRAFFIC MODEL WITH NONNEGATIVE VELOCITIES. BY INCORPORATING RECENT ANTIDIFFUSIVE TECHNIQUES FOR TRANSPORT EQUATIONS A NEW VERSION OF THESE LAGRANGIAN-ANTIDIFFUSIVE REMAP (L-AR) SCHEMES FOR THE POLYDISPERSE SEDIMENTATION MODEL IS CONSTRUCTED. THESE L-AR SCHEMES ARE SUPPORTED BY A PARTIAL ANALYSIS FOR N?=?1. THEY ARE TOTAL VARIATION DIMINISHING UNDER A SUITABLE CFL CONDITION AND THEREFORE CONVERGE TO A WEAK SOLUTION. NUMERICAL EXAMPLES ILLUSTRATE THAT THESE SCHEMES, INCLUDING A MORE ACCURATE VERSION BASED ON MUSCL EXTRAPOLATION, ARE COMPETITIVE IN ACCURACY AND EFFICIENCY WITH SEVERAL EXISTING SCHEMES.

WE STUDY A 1D SCALAR CONSERVATION LAW WHOSE NON-LOCAL FLUX HAS A SINGLE SPATIAL DISCONTINUITY. THIS MODEL IS INTENDED TO DESCRIBE TRAFFIC FLOW ON A ROAD WITH ROUGH CONDITIONS. WE APPROXIMATE THE PROBLEM THROUGH AN UPWIND-TYPE NUMERICAL SCHEME AND PROVIDE COMPACTNESS ESTIMATES FOR THE SEQUENCE OF APPROXIMATE SOLUTIONS. THEN, WE PROVE THE EXISTENCE AND THE UNIQUENESS OF ENTROPY WEAK SOLUTIONS. NUMERICAL SIMULATIONS CORROBORATE THE THEORETICAL RESULTS AND THE LIMIT MODEL AS THE KERNEL SUPPORT TENDS TO ZERO IS NUMERICALLY INVESTIGATED.

THE AIM OF THIS ARTICLE IS TO INVESTIGATE THE EXISTENCE OF TRAVELING WAVES OF A DIFFUSIVE MODEL THAT REPRESENTS THE TRANSMISSION OF A VIRUS IN A DETERMINED POPULATION COMPOSED OF THE FOLLOWING POPULATIONS: SUSCEPTIBLE ( S ) , INFECTED ( I ) , ASYMPTOMATIC ( A ) , AND RECOVERED ( R ) . AN ANALYTICAL STUDY IS PERFORMED, WHERE THE EXISTENCE OF SOLUTIONS OF TRAVELING WAVES IN A BOUNDED DOMAIN IS DEMONSTRATED. WE USE THE UPPER AND LOWER COUPLED SOLUTIONS METHOD TO ACHIEVE THIS AIM. THE EXISTENCE AND LOCAL ASYMPTOTIC STABILITY OF THE ENDEMIC ( E E ) AND DISEASE-FREE ( E 0 ) EQUILIBRIUM STATES ARE ALSO DETERMINED. THE CONSTRUCTED MODEL INCLUDES A DISCRETE-TIME DELAY THAT IS RELATED TO THE INCUBATION STAGE OF A VIRUS. WE FIND THE CRUCIAL BASIC REPRODUCTION NUMBER R 0 , WHICH DETERMINES THE LOCAL STABILITY OF THE STEADY STATES. WE PERFORM NUMERICAL SIMULATIONS OF THE MODEL IN ORDER TO PROVIDE ADDITIONAL SUPPORT TO THE THEORETICAL RESULTS AND OBSERVE THE TRAVELING WAVES. THE MODEL CAN BE USED TO STUDY THE DYNAMICS OF SARS-COV-2 AND OTHER VIRUSES WHERE THE DISEASE EVOLUTION HAS A SIMILAR BEHAVIOR.

THE PROPAGATION OF A FOREST FIRE CAN BE DESCRIBED BY A CONVECTION?DIFFUSION?REACTION PROBLEM IN TWO SPATIAL DIMENSIONS, WHERE THE UNKNOWNS ARE THE LOCAL TEMPERATURE AND THE PORTION OF FUEL CONSUMED AS FUNCTIONS OF SPATIAL POSITION AND TIME. THIS MODEL CAN BE SOLVED NUMERICALLY IN AN EFFICIENT WAY BY A LINEARLY IMPLICIT?EXPLICIT (IMEX) METHOD TO DISCRETIZE THE CONVECTION AND NONLINEAR DIFFUSION TERMS COMBINED WITH A STRANG-TYPE OPERATOR SPLITTING TO HANDLE THE REACTION TERM. THIS METHOD IS APPLIED TO SEVERAL VARIANTS OF THE MODEL WITH VARIABLE, NONLINEAR DIFFUSION FUNCTIONS, WHERE IT TURNS OUT THAT INCREASING DIFFUSIVITY (WITH RESPECT TO A GIVEN BASE CASE) SIGNIFICANTLY ENLARGES THE PORTION OF FUEL BURNT WITHIN A GIVEN TIME WHILE CHOOSING AN EQUIVALENT CONSTANT DIFFUSIVITY OR A DEGENERATE ONE PRODUCES COMPARABLE RESULTS FOR THAT QUANTITY. IN ADDITION, THE EFFECT OF SPATIAL HETEROGENEITY AS DESCRIBED BY A VARIABLE TOPOGRAPHY IS STUDIED. THE VARIABILITY OF TOPOGRAPHY INFLUENCES THE LOCAL VELOCITY AND DIRECTION OF WIND. IT IS DEMONSTRATED HOW THIS VARIABILITY AFFECTS THE DIRECTION AND SPEED OF PROPAGATION OF THE WILDFIRE AND THE LOCATION AND SIZE OF THE AREA OF FUEL CONSUMED. THE POSSIBILITY TO SOLVE THE BASE MODEL EFFICIENTLY IS UTILIZED FOR THE COMPUTATION OF SO-CALLED RISK MAPS. HERE THE RISK ASSOCIATED WITH A GIVEN POSITION IN A SUB-AREA OF THE COMPUTATIONAL DOMAIN IS QUANTIFIED BY THE RAPIDITY OF CONSUMPTION OF A GIVEN AMOUNT OF FUEL BY A FIRE STARTING IN THAT POSITION. AS A RESULT, WE OBTAIN THAT, IN COMPARISON WITH THE PLANAR CASE AND UNDER THE SAME WIND CONDITIONS, THE MODEL PREDICTS A HIGHER RISK FOR THOSE AREAS WHERE BOTH THE VARIABILITY OF TOPOGRAPHY (AS EXPRESSED BY THE GRADIENT OF ITS HEIGHT FUNCTION) AND THE WIND VELOCITY ARE INFLUENTIAL. IN GENERAL, NUMERICAL SIMULATIONS SHOW THAT IN ALL CASES THE RISK MAP WITH FOR A NON-PLANAR TOPOGRAPHY INCLUDES AREAS WITH A REDUCED RISK AS WELL AS SUCH WITH AN ENHANCED RISK AS COMPARED TO THE PLANAR CASE.

A MODEL FOR THE SPATIO-TEMPORAL EVOLUTION OF THREE BIOLOGICAL SPECIES IN A FOOD CHAIN MODEL CONSISTING OF TWO COMPETITIVE PREYS AND ONE PREDATOR WITH INTRA SPECIFIC COMPETITION IS CONSIDERED. BESIDES DIFFUSING, THE PREDATOR SPECIES MOVES TOWARD HIGHER CONCENTRATIONS OF A CHEMICAL SUBSTANCE PRODUCED BY THE PREY. THE PREY, IN TURN, MOVES AWAY FROM HIGH CONCENTRATIONS OF A SUBSTANCE SECRETED BY THE PREDATORS. THE RESULTING REACTION?DIFFUSION SYSTEM CONSISTS OF THREE PARABOLIC EQUATIONS ALONG WITH THREE ELLIPTIC EQUATIONS DESCRIBING THE DIFFUSION OF THE CHEMICAL SUBSTANCES. THE LOCAL EXISTENCE OF NONNEGATIVE SOLUTIONS IS PROVED. THEN UNIFORM ESTIMATES IN LEBESGUE SPACES ARE PROVIDED. THESE ESTIMATES LEAD TO BOUNDEDNESS AND GLOBAL WELL-POSEDNESS FOR THE SYSTEM. NUMERICAL SIMULATIONS ARE PRESENTED AND DISCUSSED.

THIS PAPER FOCUSES ON THE NUMERICAL APPROXIMATION OF A CLASS OF NON-LOCAL SYSTEMS OF CONSERVATION LAWS IN ONE SPACE DIMENSION, ARISING IN TRAFFIC MODELING, PROPOSED BY [F.A. CHIARELLO AND P. GOATIN. NON-LOCAL MULTI-CLASS TRAFFIC FLOW MODELS. NETWORKS AND HETEROGE-NEOUS MEDIA, TO APPEAR, AUG. 2018]. WE PRESENT THE MULTI-CLASS VERSION OF THE FINITE VOLUME WENO (FV-WENO) SCHEMES [C. CHALONS, P. GOATIN, AND L. M. VILLADA. HIGH-ORDER NUMERICAL SCHEMES FOR ONE-DIMENSIONAL NON-LOCAL CONSERVATION LAWS. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 40(1):A288?A305, 2018.], WITH QUADRATIC POLYNOMIAL RECONSTRUCTION IN EACH CELL TO EVALUATE THE NON-LOCAL TERMS IN ORDER TO OBTAIN HIGH-ORDER OF ACCURACY. SIMULATIONS USING FV-WENO SCHEMES FOR A MULTI-CLASS MODEL FOR AUTONOMOUS AND HUMAN-DRIVEN TRAFFIC FLOW ARE PRESENTED FOR M = 3 .

THIS PAPER FOCUSES ON THE NUMERICAL APPROXIMATION OF THE SOLUTIONS OF NONLOCAL CONSERVATION LAWS IN ONE SPACE DIMENSION. THESE EQUATIONS ARE MOTIVATED BY TWO DISTINCT APPLICATIONS, NAMELY, A TRAFFIC FLOW MODEL IN WHICH THE MEAN VELOCITY DEPENDS ON A WEIGHTED MEAN OF THE DOWNSTREAM TRAFFIC DENSITY, AND A SEDIMENTATION MODEL WHERE EITHER THE SOLID PHASE VELOCITY OR THE SOLID-FLUID RELATIVE VELOCITY DEPENDS ON THE CONCENTRATION IN A NEIGHBORHOOD. IN BOTH MODELS, THE VELOCITY IS A FUNCTION OF A CONVOLUTION PRODUCT BETWEEN THE UNKNOWN AND A KERNEL FUNCTION WITH COMPACT SUPPORT. IT TURNS OUT THAT THE SOLUTIONS OF SUCH EQUATIONS MAY EXHIBIT OSCILLATIONS THAT ARE VERY DIFFICULT TO APPROXIMATE USING CLASSICAL FIRST-ORDER NUMERICAL SCHEMES. WE PROPOSE TO DESIGN DISCONTINUOUS GALERKIN (DG) SCHEMES AND FINITE VOLUME WENO (FV-WENO) SCHEMES TO OBTAIN HIGH-ORDER APPROXIMATIONS. AS WE WILL SEE, THE DG SCHEMES GIVE THE BEST NUMERICAL RESULTS BUT THEIR CFL CONDITION IS VERY RESTRICTIVE. ON THE CONTRARY, FV-WENO SCHEMES CAN BE USED WITH LARGER TIME STEPS. WE WILL SEE THAT THE EVALUATION OF THE CONVOLUTION TERMS NECESSITATES THE USE OF QUADRATIC POLYNOMIALS RECONSTRUCTIONS IN EACH CELL IN ORDER TO OBTAIN HIGH-ORDER ACCURACY WITH THE FV-WENO APPROACH. SIMULATIONS USING DG AND FV-WENO SCHEMES ARE PRESENTED FOR BOTH APPLICATIONS.

THE NUMERICAL SOLUTION OF NONLINEAR CONVECTION-DIFFUSION EQUATIONS WITH NONLOCAL FLUX BY EXPLICIT FINITE DIFFERENCE METHODS IS COSTLY DUE TO THE LOCAL SPATIAL CONVOLUTION WITHIN THE CONVECTIVE NUMERICAL FLUX AND THE DISADVANTAGEOUS COURANT-FRIEDRICHS-LEWY (CFL) CONDITION CAUSED BY THE DIFFUSION TERM. MORE EFFICIENT NUMERICAL METHODS ARE OBTAINED BY APPLYING SECOND-ORDER IMPLICIT-EXPLICIT (IMEX) RUNGE-KUTTA TIME DISCRETIZATIONS TO AN AVAILABLE EXPLICIT SCHEME FOR SUCH MODELS IN CARRILLO ET AL. (2015) [13]. THE RESULTING IMEX-RK METHODS REQUIRE SOLVING NONLINEAR ALGEBRAIC SYSTEMS IN EVERY TIME STEP. IT IS PROVEN, FOR A GENERAL NUMBER OF SPACE DIMENSIONS, THAT THIS METHOD IS WELL DEFINED. NUMERICAL EXPERIMENTS FOR SPATIALLY TWO-DIMENSIONAL PROBLEMS MOTIVATED BY MODELS OF COLLECTIVE BEHAVIOUR ARE CONDUCTED WITH SEVERAL ALTERNATIVE CHOICES OF THE PAIR OF RUNGE-KUTTA SCHEMES DEFINING AN IMEX-RK METHOD. FOR FINE DISCRETIZATIONS, IMEX-RK METHODS TURN OUT MORE EFFICIENT IN TERMS OF REDUCTION OF ERROR VERSUS CPU TIME THAN THE ORIGINAL EXPLICIT METHOD.

NUMERICAL TECHNIQUES FOR APPROXIMATE SOLUTION OF A SYSTEM OF REACTION-DIFFUSION-CONVECTION PARTIAL DIFFERENTIAL EQUATIONS MODELING THE EVOLUTION OF TEMPERATURE AND FUEL DENSITY IN A WILDFIRE ARE PROPOSED. THESE SCHEMES COMBINE LINEARLY IMPLICIT-EXPLICIT RUNGE?KUTTA (IMEX-RK) METHODS AND STRANG-TYPE SPLITTING TECHNIQUE TO ADEQUATELY HANDLE THE NON-LINEAR PARABOLIC TERM AND THE STIFFNESS IN THE REACTIVE PART. WEIGHTED ESSENTIALLY NON-OSCILLATORY (WENO) RECONSTRUCTIONS ARE APPLIED TO THE DISCRETIZATION OF THE NONLINEAR CONVECTION TERM. EXAMPLES ARE FOCUSED ON THE APPLICATIVE PROBLEM OF DETERMINING THE WIDTH OF A FIREBREAK TO PREVENT THE PROPAGATION OF FOREST FIRES. RESULTS ILLUSTRATE THAT THE MODEL AND NUMERICAL SCHEME PROVIDE AN EFFECTIVE TOOL FOR DEFINING THAT WIDTH AND THE PARAMETERS FOR CONTROL STRATEGIES OF WILDLAND FIRES.

NONLINEAR CONVECTION?DIFFUSION EQUATIONS WITH NONLOCAL FLUX AND POSSIBLY DEGENERATE DIFFUSION ARISE IN VARIOUS CONTEXTS INCLUDING INTERACTING GASES, POROUS MEDIA FLOWS, AND COLLECTIVE BEHAVIOR IN BIOLOGY. THEIR NUMERICAL SOLUTION BY AN EXPLICIT FINITE DIFFERENCE METHOD IS COSTLY DUE TO THE NECESSITY OF DISCRETIZING A LOCAL SPATIAL CONVOLUTION FOR EACH EVALUATION OF THE CONVECTIVE NUMERICAL FLUX, AND DUE TO THE DISADVANTAGEOUS COURANT?FRIEDRICHS?LEWY (CFL) CONDITION INCURRED BY THE DIFFUSION TERM. BASED ON EXPLICIT SCHEMES FOR SUCH MODELS DEVISED IN THE STUDY OF CARRILLO ET AL. A SECOND-ORDER IMPLICIT?EXPLICIT RUNGE?KUTTA (IMEX-RK) METHOD CAN BE FORMULATED. THIS METHOD AVOIDS THE RESTRICTIVE TIME STEP LIMITATION OF EXPLICIT SCHEMES SINCE THE DIFFUSION TERM IS HANDLED IMPLICITLY, BUT ENTAILS THE NECESSITY TO SOLVE NONLINEAR ALGEBRAIC SYSTEMS IN EVERY TIME STEP. IT IS PROVEN THAT THIS METHOD IS WELL DEFINED. NUMERICAL EXPERIMENTS ILLUSTRATE THAT FOR FINE DISCRETIZATIONS IT IS MORE EFFICIENT IN TERMS OF REDUCTION OF ERROR VERSUS CENTRAL PROCESSING UNIT TIME THAN THE ORIGINAL EXPLICIT METHOD. ONE OF THE TEST CASES IS GIVEN BY A STRONGLY DEGENERATE PARABOLIC, NONLOCAL EQUATION MODELING AGGREGATION IN STUDY OF BETANCOURT ET AL. THIS MODEL CAN BE TRANSFORMED TO A LOCAL PARTIAL DIFFERENTIAL EQUATION THAT CAN BE SOLVED NUMERICALLY EASILY TO GENERATE A REFERENCE SOLUTION FOR THE IMEX-RK METHOD, BUT IS LIMITED TO ONE SPACE DIMENSION.

THIS PAPER FOCUSES ON THE NUMERICAL APPROXIMATION OF THE SOLUTIONS OF A CLASS OF NON-LOCAL SYSTEMS IN ONE SPACE DIMENSION, ARISING IN TRAFFIC MODELING. WE PROPOSE ALTERNATIVE SIMPLE SCHEMES BY SPLITTING THE NON-LOCAL CONSERVATION LAWS INTO TWO DIFFERENT EQUATIONS, NAMELY THE LAGRANGIAN AND THE REMAP STEPS. WE PROVIDE SOME PROPERTIES AND ESTIMATES RECOVERED BY APPROXIMATING THE PROBLEM WITH THE LAGRANGIAN-ANTIDIFFUSIVE REMAP (L-AR) SCHEME, AND WE PROVE THE CONVERGENCE TO WEAK SOLUTIONS IN THE SCALAR CASE. FINALLY, WE SHOW SOME NUMERICAL SIMULATIONS ILLUSTRATING THE EFFICIENCY OF THE L-AR SCHEMES IN COMPARISON WITH CLASSICAL FIRST- AND SECOND-ORDER NUMERICAL SCHEMES.

MULTISPECIES KINEMATIC FLOW MODELS WITH STRONGLY DEGENERATE DIFFUSIVE CORRECTIONS GIVE RISE TO SYSTEMS OF NONLINEAR CONVECTION-DIFFUSION EQUATIONS OF ARBITRARY SIZE. APPLICATIONS OF THESE SYSTEMS INCLUDE MODELS OF POLYDISPERSE SEDIMENTATION AND MULTICLASS TRAFFIC FLOW. IMPLICIT-EXPLICIT (IMEX) RUNGE--KUTTA (RK) METHODS ARE SUITABLE FOR THE SOLUTION OF THESE CONVECTION-DIFFUSION PROBLEMS SINCE THE STABILITY RESTRICTIONS, COMING FROM THE EXPLICITLY TREATED CONVECTIVE PART, ARE MUCH LESS SEVERE THAN THOSE THAT WOULD BE DEDUCED FROM AN EXPLICIT TREATMENT OF THE DIFFUSIVE TERM. THESE SCHEMES USUALLY COMBINE AN EXPLICIT RK SCHEME FOR THE TIME INTEGRATION OF THE CONVECTIVE PART WITH A DIAGONALLY IMPLICIT ONE FOR THE DIFFUSIVE PART. IN [R. BÜRGER, P. MULET, AND L. M. VILLADA, SIAM J. SCI. COMPUT., 35 (2013), PP. B751--B777] A SCHEME OF THIS TYPE IS PROPOSED, WHERE THE NONLINEAR AND NONSMOOTH SYSTEMS OF ALGEBRAIC EQUATIONS ARISING IN THE IMPLICIT TREATMENT OF THE DEGENERATE DIFFUSIVE PART ARE SOLVED BY SMOOTHING OF THE DIFFUSION COEFFICIENTS COMBINED WITH A NEWTON--RAPHSON METHOD WITH LINE SEARCH. THIS NONLINEARLY IMPLICIT METHOD IS ROBUST BUT ASSOCIATED WITH CONSIDERABLE EFFORT OF IMPLEMENTATION AND POSSIBLY CPU TIME. TO OVERCOME THESE SHORTCOMINGS WHILE KEEPING THE ADVANTAGEOUS STABILITY PROPERTIES OF IMEX-RK METHODS, A SECOND VARIANT OF THESE METHODS IS PROPOSED IN WHICH THE DIFFUSION TERMS ARE DISCRETIZED IN A WAY THAT MORE CAREFULLY DISTINGUISHES BETWEEN STIFF AND NONSTIFF DEPENDENCE, SUCH THAT IN EACH TIME STEP ONLY A LINEAR SYSTEM NEEDS TO BE SOLVED STILL MAINTAINING HIGH ORDER ACCURACY IN TIME, WHICH MAKES THESE METHODS MUCH SIMPLER TO IMPLEMENT. IN A SERIES OF EXAMPLES OF POLYDISPERSE SEDIMENTATION AND MULTICLASS TRAFFIC FLOW, IT IS DEMONSTRATED THAT THESE NEW LINEARLY IMPLICIT IMEX-RK SCHEMES APPROXIMATE THE SAME SOLUTIONS AS THE NONLINEARLY IMPLICIT VERSIONS, AND IN MANY CASES THESE SCHEMES ARE MORE EFFICIENT.

WE PRESENT A NON-LOCAL VERSION OF A SCALAR BALANCE LAW MODELING TRAFFIC FLOW WITH ON-RAMPS AND OFF-RAMPS. THE SOURCE TERM IS USED TO DESCRIBE THE INFLOW AND OUTPUT FLOW OVER THE ON-RAMP AND OFF-RAMPS RESPECTIVELY. WE APPROXIMATE THE PROBLEM USING AN UPWIND-TYPE NUMERICAL SCHEME AND WE PROVIDE L? AND BV ESTIMATES FOR THE SEQUENCE OF APPROXIMATE SOLUTIONS. TOGETHER WITH A DISCRETE ENTROPY INEQUALITY, WE ALSO SHOW THE WELL-POSEDNESS OF THE CONSIDERED CLASS OF SCALAR BALANCE LAWS. SOME NUMERICAL SIMULATIONS ILLUSTRATE THE BEHAVIOUR OF SOLUTIONS IN SAMPLE CASES.

A REACTION?DIFFUSION SYSTEM IS FORMULATED TO DESCRIBE THREE INTERACTING SPECIES WITHIN THE HASTINGS?POWELL (HP) FOOD CHAIN STRUCTURE WITH CHEMOTAXIS PRODUCED BY THREE CHEMICALS. WE CONSTRUCT A FINITE VOLUME (FV) SCHEME FOR THIS SYSTEM, AND IN COMBINATION WITH THE NON-NEGATIVITY AND A PRIORI ESTIMATES FOR THE DISCRETE SOLUTION, THE EXISTENCE OF A DISCRETE SOLUTION OF THE FV SCHEME IS PROVEN. IT IS SHOWN THAT THE SCHEME CONVERGES TO THE CORRESPONDING WEAK SOLUTION OF THE MODEL. THE CONVERGENCE PROOF USES TWO INGREDIENTS OF INTEREST FOR VARIOUS APPLICATIONS, NAMELY THE DISCRETE SOBOLEV EMBEDDING INEQUALITIES WITH GENERAL BOUNDARY CONDITIONS AND A SPACE?TIME COMPACTNESS ARGUMENT. FINALLY, NUMERICAL TESTS ILLUSTRATE THE MODEL AND THE BEHAVIOR OF THE FV SCHEME.